| L(s) = 1 | + 1.22·3-s + 3.08·7-s − 1.50·9-s + 3.83·11-s − 5.85·13-s − 6.54·17-s − 19-s + 3.77·21-s + 4.37·23-s − 5.50·27-s − 4.41·29-s − 0.451·31-s + 4.69·33-s − 2.49·37-s − 7.15·39-s − 1.03·41-s + 3.19·43-s − 3.70·47-s + 2.52·49-s − 7.99·51-s − 5.19·53-s − 1.22·57-s + 7.18·59-s + 5.02·61-s − 4.65·63-s − 13.1·67-s + 5.34·69-s + ⋯ |
| L(s) = 1 | + 0.705·3-s + 1.16·7-s − 0.502·9-s + 1.15·11-s − 1.62·13-s − 1.58·17-s − 0.229·19-s + 0.822·21-s + 0.911·23-s − 1.05·27-s − 0.820·29-s − 0.0810·31-s + 0.816·33-s − 0.409·37-s − 1.14·39-s − 0.162·41-s + 0.486·43-s − 0.540·47-s + 0.360·49-s − 1.11·51-s − 0.713·53-s − 0.161·57-s + 0.935·59-s + 0.643·61-s − 0.586·63-s − 1.60·67-s + 0.642·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.22T + 3T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 0.451T + 31T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 2.49T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 0.245T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.46119873489270984147938594733, −7.08798706175996321825844107230, −6.20045279715408610384943517753, −5.26389188333892663909277213355, −4.63206573340329444907955993334, −4.03074601243493776864288979693, −2.97845914929348359928696872621, −2.23343770572331742068642760893, −1.55965799822884664057354508439, 0,
1.55965799822884664057354508439, 2.23343770572331742068642760893, 2.97845914929348359928696872621, 4.03074601243493776864288979693, 4.63206573340329444907955993334, 5.26389188333892663909277213355, 6.20045279715408610384943517753, 7.08798706175996321825844107230, 7.46119873489270984147938594733
