L(s) = 1 | + 2.98·5-s − 11.7·7-s + 55.2·11-s − 5.58·13-s + 35.9·17-s − 107.·19-s − 162.·23-s − 116.·25-s + 15.2·29-s − 108.·31-s − 34.9·35-s + 415.·37-s − 57.5·41-s + 178.·43-s + 463.·47-s − 205.·49-s − 234.·53-s + 164.·55-s − 199.·59-s + 849.·61-s − 16.6·65-s − 1.04e3·67-s − 533.·71-s + 477.·73-s − 646.·77-s − 1.05e3·79-s − 1.05e3·83-s + ⋯ |
L(s) = 1 | + 0.266·5-s − 0.632·7-s + 1.51·11-s − 0.119·13-s + 0.513·17-s − 1.29·19-s − 1.47·23-s − 0.928·25-s + 0.0974·29-s − 0.628·31-s − 0.168·35-s + 1.84·37-s − 0.219·41-s + 0.631·43-s + 1.43·47-s − 0.600·49-s − 0.608·53-s + 0.403·55-s − 0.440·59-s + 1.78·61-s − 0.0317·65-s − 1.90·67-s − 0.892·71-s + 0.765·73-s − 0.957·77-s − 1.50·79-s − 1.40·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.98T + 125T^{2} \) |
| 7 | \( 1 + 11.7T + 343T^{2} \) |
| 11 | \( 1 - 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.58T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 415.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 57.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 199.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 849.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 533.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 608.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 263.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.022721528333110139380415500940, −8.104814229297837553603082291290, −7.16970911676972902100726069171, −6.18868736246125577306099254983, −5.88681194758267653414796029901, −4.31758536351591819126209831635, −3.80588618809512163133851532054, −2.49534994930354631296234623349, −1.41672979659177830710996395648, 0,
1.41672979659177830710996395648, 2.49534994930354631296234623349, 3.80588618809512163133851532054, 4.31758536351591819126209831635, 5.88681194758267653414796029901, 6.18868736246125577306099254983, 7.16970911676972902100726069171, 8.104814229297837553603082291290, 9.022721528333110139380415500940