| L(s) = 1 | − 3·5-s − 3·9-s + 5·11-s + 4·13-s − 2·17-s − 19-s − 9·23-s + 4·25-s + 2·29-s + 6·31-s − 8·37-s + 6·41-s − 5·43-s + 9·45-s + 9·47-s + 2·53-s − 15·55-s + 5·61-s − 12·65-s − 12·67-s + 10·71-s − 73-s + 4·79-s + 9·81-s + 3·83-s + 6·85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 9-s + 1.50·11-s + 1.10·13-s − 0.485·17-s − 0.229·19-s − 1.87·23-s + 4/5·25-s + 0.371·29-s + 1.07·31-s − 1.31·37-s + 0.937·41-s − 0.762·43-s + 1.34·45-s + 1.31·47-s + 0.274·53-s − 2.02·55-s + 0.640·61-s − 1.48·65-s − 1.46·67-s + 1.18·71-s − 0.117·73-s + 0.450·79-s + 81-s + 0.329·83-s + 0.650·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.73374565238068634664688487510, −6.68942106506740032952349136898, −6.33064813597026693955922550753, −5.54009558320851669712767939146, −4.39517278576693843070112357545, −3.91895960520322868834353250014, −3.41488324415626181830727069133, −2.29626040618375194131487539022, −1.11689201761758240842719715202, 0,
1.11689201761758240842719715202, 2.29626040618375194131487539022, 3.41488324415626181830727069133, 3.91895960520322868834353250014, 4.39517278576693843070112357545, 5.54009558320851669712767939146, 6.33064813597026693955922550753, 6.68942106506740032952349136898, 7.73374565238068634664688487510
