| L(s) = 1 | − 0.414·2-s − 1.82·4-s + 5-s − 4.82·7-s + 1.58·8-s − 0.414·10-s + 11-s + 5.65·13-s + 1.99·14-s + 3·16-s + 6.82·17-s − 1.17·19-s − 1.82·20-s − 0.414·22-s + 4·23-s + 25-s − 2.34·26-s + 8.82·28-s − 0.828·29-s − 4.41·32-s − 2.82·34-s − 4.82·35-s + 0.343·37-s + 0.485·38-s + 1.58·40-s + 0.828·41-s − 3.17·43-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.447·5-s − 1.82·7-s + 0.560·8-s − 0.130·10-s + 0.301·11-s + 1.56·13-s + 0.534·14-s + 0.750·16-s + 1.65·17-s − 0.268·19-s − 0.408·20-s − 0.0883·22-s + 0.834·23-s + 0.200·25-s − 0.459·26-s + 1.66·28-s − 0.153·29-s − 0.780·32-s − 0.485·34-s − 0.816·35-s + 0.0564·37-s + 0.0787·38-s + 0.250·40-s + 0.129·41-s − 0.483·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9657007276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9657007276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−10.51843968414512898332519903296, −10.00169364499106846710884884196, −9.169724094385808836303108789430, −8.602383164944062514213902740200, −7.29379222370783295255333182248, −6.20289516130001992071945017691, −5.52114127546218421109580391860, −3.92490376183820598138637798674, −3.18629491234324629698620261642, −0.992081852202220186506946909778,
0.992081852202220186506946909778, 3.18629491234324629698620261642, 3.92490376183820598138637798674, 5.52114127546218421109580391860, 6.20289516130001992071945017691, 7.29379222370783295255333182248, 8.602383164944062514213902740200, 9.169724094385808836303108789430, 10.00169364499106846710884884196, 10.51843968414512898332519903296
