| L(s) = 1 | − 16·9-s + 100·19-s + 228·31-s + 218·49-s + 124·61-s − 152·79-s + 16·81-s − 124·109-s + 960·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.56e3·169-s − 1.60e3·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.77·9-s + 5.26·19-s + 7.35·31-s + 4.44·49-s + 2.03·61-s − 1.92·79-s + 0.197·81-s − 1.13·109-s + 7.93·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 9.24·169-s − 9.35·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(255.3507493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(255.3507493\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 16 T^{2} + 80 p T^{4} + 226 p^{2} T^{6} + 80 p^{5} T^{8} + 16 p^{8} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 109 T^{2} + 5195 T^{4} - 211774 T^{6} + 5195 p^{4} T^{8} - 109 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 - 480 T^{2} + 115680 T^{4} - 17321010 T^{6} + 115680 p^{4} T^{8} - 480 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 13 | \( ( 1 - 781 T^{2} + 285971 T^{4} - 61407934 T^{6} + 285971 p^{4} T^{8} - 781 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 + 560 T^{2} + 177744 T^{4} + 48032482 T^{6} + 177744 p^{4} T^{8} + 560 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 19 | \( ( 1 - 25 T + 878 T^{2} - 17173 T^{3} + 878 p^{2} T^{4} - 25 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 23 | \( ( 1 + 494 T^{2} + 512415 T^{4} + 138668068 T^{6} + 512415 p^{4} T^{8} + 494 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 2430 T^{2} + 3408879 T^{4} - 3448393668 T^{6} + 3408879 p^{4} T^{8} - 2430 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 - 57 T + 3171 T^{2} - 107750 T^{3} + 3171 p^{2} T^{4} - 57 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 37 | \( ( 1 - 3122 T^{2} + 8852495 T^{4} - 12809790428 T^{6} + 8852495 p^{4} T^{8} - 3122 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 5280 T^{2} + 10814208 T^{4} - 15995121410 T^{6} + 10814208 p^{4} T^{8} - 5280 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 - 5029 T^{2} + 15737555 T^{4} - 32898405646 T^{6} + 15737555 p^{4} T^{8} - 5029 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 - 4050 T^{2} + 19404543 T^{4} - 40831163100 T^{6} + 19404543 p^{4} T^{8} - 4050 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 + 7134 T^{2} + 37832847 T^{4} + 119639616644 T^{6} + 37832847 p^{4} T^{8} + 7134 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 3830 T^{2} + 26455071 T^{4} - 83897306356 T^{6} + 26455071 p^{4} T^{8} - 3830 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 - 31 T + 6875 T^{2} - 113038 T^{3} + 6875 p^{2} T^{4} - 31 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 67 | \( ( 1 - 11403 T^{2} + 96392358 T^{4} - 481205427487 T^{6} + 96392358 p^{4} T^{8} - 11403 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 9646 T^{2} + 2370655 T^{4} + 226031091292 T^{6} + 2370655 p^{4} T^{8} - 9646 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 - 19824 T^{2} + 211626960 T^{4} - 1387743416606 T^{6} + 211626960 p^{4} T^{8} - 19824 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 + 38 T + 4707 T^{2} + 844396 T^{3} + 4707 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 83 | \( ( 1 + 36128 T^{2} + 6853824 p T^{4} + 5062587230386 T^{6} + 6853824 p^{5} T^{8} + 36128 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 34048 T^{2} + 531935104 T^{4} - 5142643795106 T^{6} + 531935104 p^{4} T^{8} - 34048 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 - 17325 T^{2} + 208459395 T^{4} - 2144644177822 T^{6} + 208459395 p^{4} T^{8} - 17325 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.83415938534910400794249922916, −2.82239992628823314944857826065, −2.75754283063273541295135437001, −2.61955850508013348475784256792, −2.59907550272556817503506482277, −2.49522732202487604926187570209, −2.46313391528711880785684225870, −2.13435371618415339177948181483, −2.02319626438183584861793511163, −1.90438523740037971923781564192, −1.85971577623307949472938704929, −1.71337650613985413659556444419, −1.58337627090172641696944557789, −1.56431607400577430531435444648, −1.49490905993385134060603054116, −1.27055820101116393647253637701, −0.970188433539684069330006500584, −0.866654411415604036710262779516, −0.77355512641255487776323242848, −0.72217219690324084355948076517, −0.71515314070457400625853468044, −0.69654892830893169624705314828, −0.46760084898397587065397776435, −0.43068518991225412591775949330, −0.42677612076929038535500418713,
0.42677612076929038535500418713, 0.43068518991225412591775949330, 0.46760084898397587065397776435, 0.69654892830893169624705314828, 0.71515314070457400625853468044, 0.72217219690324084355948076517, 0.77355512641255487776323242848, 0.866654411415604036710262779516, 0.970188433539684069330006500584, 1.27055820101116393647253637701, 1.49490905993385134060603054116, 1.56431607400577430531435444648, 1.58337627090172641696944557789, 1.71337650613985413659556444419, 1.85971577623307949472938704929, 1.90438523740037971923781564192, 2.02319626438183584861793511163, 2.13435371618415339177948181483, 2.46313391528711880785684225870, 2.49522732202487604926187570209, 2.59907550272556817503506482277, 2.61955850508013348475784256792, 2.75754283063273541295135437001, 2.82239992628823314944857826065, 2.83415938534910400794249922916
Plot not available for L-functions of degree greater than 10.
