| L(s) = 1 | − 154·7-s + 942·19-s + 262·25-s − 978·31-s − 4.08e3·37-s + 6.89e3·43-s + 1.29e4·49-s + 1.44e4·61-s + 1.00e4·67-s − 1.37e4·73-s + 470·79-s − 3.35e4·103-s − 1.87e4·109-s + 1.11e4·121-s + 127-s + 131-s − 1.45e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.47e4·169-s + 173-s − 4.03e4·175-s + ⋯ |
| L(s) = 1 | − 3.14·7-s + 2.60·19-s + 0.419·25-s − 1.01·31-s − 2.98·37-s + 3.72·43-s + 5.40·49-s + 3.88·61-s + 2.24·67-s − 2.57·73-s + 0.0753·79-s − 3.16·103-s − 1.57·109-s + 0.760·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 8.20·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.31·169-s + 3.34e−5·173-s − 1.31·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9985007322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9985007322\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 11 p T + p^{4} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 262 T^{2} - 321981 T^{4} - 262 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 11138 T^{2} - 90303837 T^{4} - 11138 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 47375 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 165530 T^{2} + 20424423459 T^{4} + 165530 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 471 T + 204268 T^{2} - 471 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 446282 T^{2} + 120856638243 T^{4} - 446282 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 761378 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 489 T + 1003228 T^{2} + 489 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2041 T + 2291520 T^{2} + 2041 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3903650 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1723 T + p^{4} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 9213530 T^{2} + 61077848399139 T^{4} + 9213530 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 8391382 T^{2} + 8155601458563 T^{4} + 8391382 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 15937606 T^{2} + 107176847406915 T^{4} - 15937606 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 7224 T + 31241233 T^{2} - 7224 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 5035 T + 5200104 T^{2} - 5035 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 20051638 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6861 T + 44089348 T^{2} + 6861 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 235 T - 38894856 T^{2} - 235 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 89653370 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 83752126 T^{2} + 3077829803817795 T^{4} - 83752126 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 134826050 T^{2} + p^{8} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.025222115887343070113502366067, −7.77856740405950036427768311144, −7.32704253603927251404519761825, −7.23039631380683380493951400212, −7.04749137353066155272624019681, −6.72543255213108602338513352796, −6.67938531509642332121423845533, −6.28410962104981719000533868660, −5.73864947160201079211126197080, −5.66880082774725530714005401976, −5.46864982560129863845302462193, −5.31752907593121548705693461916, −4.85464825134137838885516478022, −4.15036677928769054394629389541, −4.08449581177526765311095646616, −3.56577205119160332571877964423, −3.53306139285255907947221442323, −3.22784267486652693044443383803, −2.77209353053252331410567693585, −2.58591714648692510514411710354, −2.19813455405074985759085845003, −1.49686689510850400725032273771, −0.936292391617819547035230239902, −0.71868139123044675092260002996, −0.19858853821725810829426556702,
0.19858853821725810829426556702, 0.71868139123044675092260002996, 0.936292391617819547035230239902, 1.49686689510850400725032273771, 2.19813455405074985759085845003, 2.58591714648692510514411710354, 2.77209353053252331410567693585, 3.22784267486652693044443383803, 3.53306139285255907947221442323, 3.56577205119160332571877964423, 4.08449581177526765311095646616, 4.15036677928769054394629389541, 4.85464825134137838885516478022, 5.31752907593121548705693461916, 5.46864982560129863845302462193, 5.66880082774725530714005401976, 5.73864947160201079211126197080, 6.28410962104981719000533868660, 6.67938531509642332121423845533, 6.72543255213108602338513352796, 7.04749137353066155272624019681, 7.23039631380683380493951400212, 7.32704253603927251404519761825, 7.77856740405950036427768311144, 8.025222115887343070113502366067
