| L(s) = 1 | + 2-s − 1.03·3-s + 4-s + 0.118·5-s − 1.03·6-s + 2.13·7-s + 8-s − 1.92·9-s + 0.118·10-s + 5.30·11-s − 1.03·12-s + 0.953·13-s + 2.13·14-s − 0.123·15-s + 16-s − 1.40·17-s − 1.92·18-s − 19-s + 0.118·20-s − 2.21·21-s + 5.30·22-s − 7.01·23-s − 1.03·24-s − 4.98·25-s + 0.953·26-s + 5.10·27-s + 2.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.598·3-s + 0.5·4-s + 0.0531·5-s − 0.423·6-s + 0.806·7-s + 0.353·8-s − 0.642·9-s + 0.0375·10-s + 1.59·11-s − 0.299·12-s + 0.264·13-s + 0.570·14-s − 0.0317·15-s + 0.250·16-s − 0.341·17-s − 0.454·18-s − 0.229·19-s + 0.0265·20-s − 0.482·21-s + 1.13·22-s − 1.46·23-s − 0.211·24-s − 0.997·25-s + 0.186·26-s + 0.982·27-s + 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.082388616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082388616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.03T + 3T^{2} \) |
| 5 | \( 1 - 0.118T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 0.953T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 23 | \( 1 + 7.01T + 23T^{2} \) |
| 29 | \( 1 + 3.22T + 29T^{2} \) |
| 31 | \( 1 - 5.86T + 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 41 | \( 1 - 6.08T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 - 8.99T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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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
−7.968248464918044616332602389221, −6.86673564897997314336112450965, −6.10420836838673270004907306628, −5.98698595571795966952450930653, −5.03402137073591851813822816152, −4.15794757612550156636474685825, −3.92847014787813130684720873475, −2.64039603555761322898197604050, −1.84233272033713333202056703182, −0.826280832445870339785328152178,
0.826280832445870339785328152178, 1.84233272033713333202056703182, 2.64039603555761322898197604050, 3.92847014787813130684720873475, 4.15794757612550156636474685825, 5.03402137073591851813822816152, 5.98698595571795966952450930653, 6.10420836838673270004907306628, 6.86673564897997314336112450965, 7.968248464918044616332602389221
