| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 4.09·13-s − 14-s + 16-s − 7.87·17-s − 0.219·19-s + 20-s − 22-s − 2.89·23-s + 25-s + 4.09·26-s − 28-s − 9.20·29-s + 32-s − 7.87·34-s − 35-s − 8.31·37-s − 0.219·38-s + 40-s − 7.42·41-s + 2.98·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.13·13-s − 0.267·14-s + 0.250·16-s − 1.91·17-s − 0.0504·19-s + 0.223·20-s − 0.213·22-s − 0.602·23-s + 0.200·25-s + 0.803·26-s − 0.188·28-s − 1.70·29-s + 0.176·32-s − 1.35·34-s − 0.169·35-s − 1.36·37-s − 0.0356·38-s + 0.158·40-s − 1.15·41-s + 0.455·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 7.87T + 17T^{2} \) |
| 19 | \( 1 + 0.219T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 + 9.20T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 - 0.987T + 61T^{2} \) |
| 67 | \( 1 + 6.67T + 67T^{2} \) |
| 71 | \( 1 + 4.21T + 71T^{2} \) |
| 73 | \( 1 + 7.78T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + 5.42T + 83T^{2} \) |
| 89 | \( 1 - 0.0978T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−7.29985202080248611945799813591, −6.83650015239775037156601699236, −6.05639059859206958428626481538, −5.62119223331644699582623609091, −4.71157664064781647726304194961, −3.95898432363682501830214545106, −3.30708670697233897510091212473, −2.27763099619327749482762829578, −1.63315868795586370433664844648, 0,
1.63315868795586370433664844648, 2.27763099619327749482762829578, 3.30708670697233897510091212473, 3.95898432363682501830214545106, 4.71157664064781647726304194961, 5.62119223331644699582623609091, 6.05639059859206958428626481538, 6.83650015239775037156601699236, 7.29985202080248611945799813591
