| L(s) = 1 | + 2-s + 3.07·3-s + 4-s + 3.07·6-s + 3.69·7-s + 8-s + 6.43·9-s − 1.06·11-s + 3.07·12-s + 3.69·14-s + 16-s − 2.79·17-s + 6.43·18-s − 2.19·19-s + 11.3·21-s − 1.06·22-s − 1.13·23-s + 3.07·24-s + 10.5·27-s + 3.69·28-s − 2.73·29-s + 4.20·31-s + 32-s − 3.27·33-s − 2.79·34-s + 6.43·36-s + 10.3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 1.25·6-s + 1.39·7-s + 0.353·8-s + 2.14·9-s − 0.321·11-s + 0.886·12-s + 0.987·14-s + 0.250·16-s − 0.678·17-s + 1.51·18-s − 0.504·19-s + 2.47·21-s − 0.227·22-s − 0.235·23-s + 0.626·24-s + 2.02·27-s + 0.698·28-s − 0.508·29-s + 0.754·31-s + 0.176·32-s − 0.570·33-s − 0.480·34-s + 1.07·36-s + 1.69·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.976798683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.976798683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 + 1.13T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 8.18T + 83T^{2} \) |
| 89 | \( 1 - 8.80T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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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
−7.933868713404798367783373204325, −7.27579967550161156595633242124, −6.56266647461784934637976568463, −5.54138289610715459089296825349, −4.67618447039304367239312371663, −4.24489859431012517692679043923, −3.52876628533581606713286052941, −2.50763766101011384785971447932, −2.19245748169349060747345960695, −1.26780077634953221649012663837,
1.26780077634953221649012663837, 2.19245748169349060747345960695, 2.50763766101011384785971447932, 3.52876628533581606713286052941, 4.24489859431012517692679043923, 4.67618447039304367239312371663, 5.54138289610715459089296825349, 6.56266647461784934637976568463, 7.27579967550161156595633242124, 7.933868713404798367783373204325
