| L(s) = 1 | − 2·3-s − 5-s + 9-s − 3·11-s − 4·13-s + 2·15-s − 2·17-s − 19-s − 7·23-s − 4·25-s + 4·27-s + 2·29-s − 6·31-s + 6·33-s − 10·37-s + 8·39-s − 8·41-s + 7·43-s − 45-s − 9·47-s + 4·51-s + 6·53-s + 3·55-s + 2·57-s − 14·59-s − 5·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.516·15-s − 0.485·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.769·27-s + 0.371·29-s − 1.07·31-s + 1.04·33-s − 1.64·37-s + 1.28·39-s − 1.24·41-s + 1.06·43-s − 0.149·45-s − 1.31·47-s + 0.560·51-s + 0.824·53-s + 0.404·55-s + 0.264·57-s − 1.82·59-s − 0.640·61-s + 0.496·65-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 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.06774048811333985507089587751, −6.53561654452187066198006821635, −5.62083942671028002016071204175, −5.23204765276196781950320505832, −4.49838089521752057038977040861, −3.68898265180486804474045985583, −2.62822660231787178083591210437, −1.75386820755316026235494309751, 0, 0,
1.75386820755316026235494309751, 2.62822660231787178083591210437, 3.68898265180486804474045985583, 4.49838089521752057038977040861, 5.23204765276196781950320505832, 5.62083942671028002016071204175, 6.53561654452187066198006821635, 7.06774048811333985507089587751
