| L(s) = 1 | + 2.65·2-s + 2.62·3-s + 5.06·4-s + 6.97·6-s + 2.51·7-s + 8.14·8-s + 3.88·9-s − 6.29·11-s + 13.2·12-s − 0.699·13-s + 6.68·14-s + 11.5·16-s − 4.58·17-s + 10.3·18-s + 7.23·19-s + 6.59·21-s − 16.7·22-s + 4.43·23-s + 21.3·24-s − 1.85·26-s + 2.31·27-s + 12.7·28-s + 3.16·29-s + 6.86·31-s + 14.3·32-s − 16.5·33-s − 12.1·34-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.51·3-s + 2.53·4-s + 2.84·6-s + 0.950·7-s + 2.87·8-s + 1.29·9-s − 1.89·11-s + 3.83·12-s − 0.193·13-s + 1.78·14-s + 2.87·16-s − 1.11·17-s + 2.43·18-s + 1.65·19-s + 1.43·21-s − 3.56·22-s + 0.925·23-s + 4.36·24-s − 0.364·26-s + 0.446·27-s + 2.40·28-s + 0.588·29-s + 1.23·31-s + 2.53·32-s − 2.87·33-s − 2.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.46714715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.46714715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 + 0.699T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 - 8.65T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.023200880047650705008678708880, −7.28757260117470474847432363371, −6.67647202264487653329939568637, −5.52953602969219089709141244107, −4.87026795360089994250252303433, −4.62993010864737999883828244693, −3.43357690252667978815990188159, −2.92303820729987177561941174861, −2.40977377971872472677672101068, −1.55787842321387664506307222485,
1.55787842321387664506307222485, 2.40977377971872472677672101068, 2.92303820729987177561941174861, 3.43357690252667978815990188159, 4.62993010864737999883828244693, 4.87026795360089994250252303433, 5.52953602969219089709141244107, 6.67647202264487653329939568637, 7.28757260117470474847432363371, 8.023200880047650705008678708880
