| L(s) = 1 | − 4·2-s + 16·4-s − 119.·7-s − 64·8-s − 263.·11-s + 851.·13-s + 478.·14-s + 256·16-s − 1.28e3·17-s + 2.06e3·19-s + 1.05e3·22-s − 55.5·23-s − 3.40e3·26-s − 1.91e3·28-s + 5.98e3·29-s + 4.78e3·31-s − 1.02e3·32-s + 5.14e3·34-s − 1.21e4·37-s − 8.24e3·38-s − 1.85e4·41-s − 2.18e3·43-s − 4.21e3·44-s + 222.·46-s + 5.59e3·47-s − 2.47e3·49-s + 1.36e4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.923·7-s − 0.353·8-s − 0.657·11-s + 1.39·13-s + 0.652·14-s + 0.250·16-s − 1.08·17-s + 1.30·19-s + 0.464·22-s − 0.0218·23-s − 0.987·26-s − 0.461·28-s + 1.32·29-s + 0.893·31-s − 0.176·32-s + 0.763·34-s − 1.45·37-s − 0.925·38-s − 1.71·41-s − 0.180·43-s − 0.328·44-s + 0.0154·46-s + 0.369·47-s − 0.147·49-s + 0.698·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 119.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 263.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 851.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 55.5T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.64e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.27e4T + 8.58e9T^{2} \) |
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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
−9.911755383800277343776432842060, −8.848290827137713978174912253748, −8.256385384360324322744808016710, −6.98831381067407799977136733548, −6.35351662166181708945168001601, −5.19401456646129692521167591079, −3.65637276813871532229859876817, −2.69337983956829755186121617241, −1.22169472352998555175481954241, 0,
1.22169472352998555175481954241, 2.69337983956829755186121617241, 3.65637276813871532229859876817, 5.19401456646129692521167591079, 6.35351662166181708945168001601, 6.98831381067407799977136733548, 8.256385384360324322744808016710, 8.848290827137713978174912253748, 9.911755383800277343776432842060
