| L(s) = 1 | + 4·2-s + 3.71·3-s + 16·4-s + 14.8·6-s + 10.2·7-s + 64·8-s − 229.·9-s + 197.·11-s + 59.4·12-s − 169·13-s + 41.0·14-s + 256·16-s + 949.·17-s − 916.·18-s + 2.23e3·19-s + 38.1·21-s + 789.·22-s + 367.·23-s + 237.·24-s − 676·26-s − 1.75e3·27-s + 164.·28-s − 6.60e3·29-s − 9.91e3·31-s + 1.02e3·32-s + 733.·33-s + 3.79e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.238·3-s + 0.5·4-s + 0.168·6-s + 0.0791·7-s + 0.353·8-s − 0.943·9-s + 0.491·11-s + 0.119·12-s − 0.277·13-s + 0.0559·14-s + 0.250·16-s + 0.797·17-s − 0.666·18-s + 1.41·19-s + 0.0188·21-s + 0.347·22-s + 0.145·23-s + 0.0843·24-s − 0.196·26-s − 0.463·27-s + 0.0395·28-s − 1.45·29-s − 1.85·31-s + 0.176·32-s + 0.117·33-s + 0.563·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.923989326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.923989326\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 - 3.71T + 243T^{2} \) |
| 7 | \( 1 - 10.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 197.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 949.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 367.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.05e4T + 8.58e9T^{2} \) |
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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
−9.622694648177122393025529204647, −9.065973635785819657636362463888, −7.76159839654417269503164052501, −7.25715286287775521038886666802, −5.80399389139785005565484171032, −5.47222337540423347793952400417, −4.07525763361489831936597206476, −3.24190532139064035248684307828, −2.22063973322207767431042394291, −0.846448305910342452938626606605,
0.846448305910342452938626606605, 2.22063973322207767431042394291, 3.24190532139064035248684307828, 4.07525763361489831936597206476, 5.47222337540423347793952400417, 5.80399389139785005565484171032, 7.25715286287775521038886666802, 7.76159839654417269503164052501, 9.065973635785819657636362463888, 9.622694648177122393025529204647
