| L(s) = 1 | − 4·2-s + 3.91·3-s + 16·4-s − 15.6·6-s − 49·7-s − 64·8-s − 227.·9-s + 337.·11-s + 62.6·12-s − 24.3·13-s + 196·14-s + 256·16-s − 1.44e3·17-s + 910.·18-s − 1.19e3·19-s − 191.·21-s − 1.34e3·22-s + 2.24e3·23-s − 250.·24-s + 97.3·26-s − 1.84e3·27-s − 784·28-s + 1.18e3·29-s + 1.04e4·31-s − 1.02e3·32-s + 1.32e3·33-s + 5.78e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.251·3-s + 0.5·4-s − 0.177·6-s − 0.377·7-s − 0.353·8-s − 0.936·9-s + 0.840·11-s + 0.125·12-s − 0.0399·13-s + 0.267·14-s + 0.250·16-s − 1.21·17-s + 0.662·18-s − 0.760·19-s − 0.0950·21-s − 0.594·22-s + 0.885·23-s − 0.0888·24-s + 0.0282·26-s − 0.486·27-s − 0.188·28-s + 0.261·29-s + 1.95·31-s − 0.176·32-s + 0.211·33-s + 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.210361909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210361909\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 3.91T + 243T^{2} \) |
| 11 | \( 1 - 337.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 24.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.19e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.24e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.04e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 825.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.61e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 369.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.15e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.36e5T + 8.58e9T^{2} \) |
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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
−10.65493272607027725867980467312, −9.514805884785053564525563840775, −8.836120366176362827822048449689, −8.120161612033376270277395357105, −6.76932909316381690827380987096, −6.19056011276703677548736759392, −4.64249938504451532075157684393, −3.24315053552903818376727085993, −2.17050523979528427168247728594, −0.63968062229248212320565641842,
0.63968062229248212320565641842, 2.17050523979528427168247728594, 3.24315053552903818376727085993, 4.64249938504451532075157684393, 6.19056011276703677548736759392, 6.76932909316381690827380987096, 8.120161612033376270277395357105, 8.836120366176362827822048449689, 9.514805884785053564525563840775, 10.65493272607027725867980467312
