| L(s) = 1 | − 1.71·2-s − 0.109·3-s + 0.928·4-s + 0.187·6-s − 4.34·7-s + 1.83·8-s − 2.98·9-s + 5.20·11-s − 0.101·12-s + 4.88·13-s + 7.43·14-s − 4.99·16-s + 5.11·18-s − 1.64·19-s + 0.475·21-s − 8.90·22-s + 8.39·23-s − 0.200·24-s − 8.35·26-s + 0.656·27-s − 4.03·28-s − 0.485·29-s − 6.30·31-s + 4.88·32-s − 0.570·33-s − 2.77·36-s + 0.930·37-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.0632·3-s + 0.464·4-s + 0.0765·6-s − 1.64·7-s + 0.648·8-s − 0.995·9-s + 1.56·11-s − 0.0293·12-s + 1.35·13-s + 1.98·14-s − 1.24·16-s + 1.20·18-s − 0.376·19-s + 0.103·21-s − 1.89·22-s + 1.75·23-s − 0.0410·24-s − 1.63·26-s + 0.126·27-s − 0.761·28-s − 0.0901·29-s − 1.13·31-s + 0.862·32-s − 0.0993·33-s − 0.462·36-s + 0.152·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7258056080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7258056080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 + 0.109T + 3T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 + 0.485T + 29T^{2} \) |
| 31 | \( 1 + 6.30T + 31T^{2} \) |
| 37 | \( 1 - 0.930T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 0.201T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−8.237126103703980029671307460678, −7.09595349185593582698055020124, −6.70720938893034373383489259557, −6.11853713194842617220245430065, −5.31712890512299386744742192234, −4.01733866600204630814665375195, −3.56308756759388334710555430370, −2.64644956612642986094152809872, −1.38402132704948787420167731356, −0.57098637886890463833698437683,
0.57098637886890463833698437683, 1.38402132704948787420167731356, 2.64644956612642986094152809872, 3.56308756759388334710555430370, 4.01733866600204630814665375195, 5.31712890512299386744742192234, 6.11853713194842617220245430065, 6.70720938893034373383489259557, 7.09595349185593582698055020124, 8.237126103703980029671307460678
