| L(s) = 1 | + 4·9-s + 24·25-s − 48·37-s + 16·81-s − 144·109-s − 176·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 120·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 96·225-s + 227-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 24/5·25-s − 7.89·37-s + 16/9·81-s − 13.7·109-s − 16·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 9.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 32/5·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2261633911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2261633911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 6 T^{2} + 38 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 + p T^{2} )^{16} \) |
| 13 | \( ( 1 + 30 T^{2} + 542 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 12 T^{2} + 530 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 10 T^{2} + 558 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 8 T^{2} + 318 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 32 T^{2} + 1182 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 10 T + p T^{2} )^{8}( 1 + 10 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{8} \) |
| 41 | \( ( 1 - 108 T^{2} + 5522 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 40 T^{2} + 3342 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 20 T^{2} + 1494 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 - 100 T^{2} + 5094 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 110 T^{2} + 8286 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 + 174 T^{2} + 13982 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 16 T + p T^{2} )^{8}( 1 + 16 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 32 T^{2} + 3534 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 48 T^{2} + 9890 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 + 38 T^{2} - 1170 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 300 T^{2} + 38258 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.26376648449295030519813061636, −2.18163599066935038818789561074, −2.14398339864063607018332123047, −2.11992196419696910433883540208, −1.95500393922573853964274194191, −1.69708988789374807302235283943, −1.63441506154747344285367609335, −1.52851179465473193697867683737, −1.50507522720360483509853255266, −1.47500823857382520322200384859, −1.47157019654962619058528487150, −1.38366835152986026566625455895, −1.33574688500034873458905242142, −1.30787914276028695787481620968, −1.24321048596786767867957207934, −1.20942598145649040012859361967, −1.05266019353768213869728442869, −0.977655826003258989203126608043, −0.900020064271311023397419323673, −0.69173339477106989740752220423, −0.42421961734103385154340242545, −0.35053350011636258480582990773, −0.21415218180996118973413405457, −0.15224976525002371117795906877, −0.05127888894417966603979022133,
0.05127888894417966603979022133, 0.15224976525002371117795906877, 0.21415218180996118973413405457, 0.35053350011636258480582990773, 0.42421961734103385154340242545, 0.69173339477106989740752220423, 0.900020064271311023397419323673, 0.977655826003258989203126608043, 1.05266019353768213869728442869, 1.20942598145649040012859361967, 1.24321048596786767867957207934, 1.30787914276028695787481620968, 1.33574688500034873458905242142, 1.38366835152986026566625455895, 1.47157019654962619058528487150, 1.47500823857382520322200384859, 1.50507522720360483509853255266, 1.52851179465473193697867683737, 1.63441506154747344285367609335, 1.69708988789374807302235283943, 1.95500393922573853964274194191, 2.11992196419696910433883540208, 2.14398339864063607018332123047, 2.18163599066935038818789561074, 2.26376648449295030519813061636
Plot not available for L-functions of degree greater than 10.
