| L(s) = 1 | + 1.19·3-s − 1.56·5-s − 3.07·7-s − 1.56·9-s + 5.47·11-s − 1.87·15-s − 2.12·17-s + 7.34·19-s − 3.68·21-s + 3.59·23-s − 2.56·25-s − 5.47·27-s − 5·29-s − 6.67·31-s + 6.56·33-s + 4.79·35-s + 6.12·37-s − 4.12·41-s + 0.673·43-s + 2.43·45-s + 0.525·47-s + 2.43·49-s − 2.54·51-s + 1.56·53-s − 8.54·55-s + 8.80·57-s + 10.2·59-s + ⋯ |
| L(s) = 1 | + 0.692·3-s − 0.698·5-s − 1.16·7-s − 0.520·9-s + 1.64·11-s − 0.483·15-s − 0.514·17-s + 1.68·19-s − 0.804·21-s + 0.750·23-s − 0.512·25-s − 1.05·27-s − 0.928·29-s − 1.19·31-s + 1.14·33-s + 0.810·35-s + 1.00·37-s − 0.643·41-s + 0.102·43-s + 0.363·45-s + 0.0767·47-s + 0.348·49-s − 0.356·51-s + 0.214·53-s − 1.15·55-s + 1.16·57-s + 1.33·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732248742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732248742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 0.673T + 43T^{2} \) |
| 47 | \( 1 - 0.525T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 7.34T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.253689398655368350759765156239, −7.34418748608151741458011174678, −6.95726595943047270962149833585, −6.07378945709666261121269381526, −5.37638629793343289107260941343, −4.14475027816488663214398999224, −3.54276880073823827291097436863, −3.14373972485910045100165482621, −1.98110309694714902692313067561, −0.67253584796936636256325940460,
0.67253584796936636256325940460, 1.98110309694714902692313067561, 3.14373972485910045100165482621, 3.54276880073823827291097436863, 4.14475027816488663214398999224, 5.37638629793343289107260941343, 6.07378945709666261121269381526, 6.95726595943047270962149833585, 7.34418748608151741458011174678, 8.253689398655368350759765156239
