| L(s) = 1 | + 3·2-s + 4-s − 5·5-s − 21·8-s − 15·10-s + 60·11-s − 38·13-s − 71·16-s + 84·17-s − 110·19-s − 5·20-s + 180·22-s + 120·23-s + 25·25-s − 114·26-s + 162·29-s − 236·31-s − 45·32-s + 252·34-s − 376·37-s − 330·38-s + 105·40-s + 126·41-s − 34·43-s + 60·44-s + 360·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.447·5-s − 0.928·8-s − 0.474·10-s + 1.64·11-s − 0.810·13-s − 1.10·16-s + 1.19·17-s − 1.32·19-s − 0.0559·20-s + 1.74·22-s + 1.08·23-s + 1/5·25-s − 0.859·26-s + 1.03·29-s − 1.36·31-s − 0.248·32-s + 1.27·34-s − 1.67·37-s − 1.40·38-s + 0.415·40-s + 0.479·41-s − 0.120·43-s + 0.205·44-s + 1.15·46-s + 0.0186·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 6 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 880 T + p^{3} T^{2} \) |
| 67 | \( 1 + 826 T + p^{3} T^{2} \) |
| 71 | \( 1 + 666 T + p^{3} T^{2} \) |
| 73 | \( 1 - 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 + 792 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1002 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1442 T + p^{3} T^{2} \) |
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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
−8.488847613102835692075792235747, −7.20683429850699212519306839750, −6.72297468342721334061609536728, −5.77055094322008016046467237423, −5.01311401317715345645308051683, −4.16473400889556679375440605053, −3.63169567894509869023591046548, −2.68511678131630060941818174849, −1.31557550760411030477401019365, 0,
1.31557550760411030477401019365, 2.68511678131630060941818174849, 3.63169567894509869023591046548, 4.16473400889556679375440605053, 5.01311401317715345645308051683, 5.77055094322008016046467237423, 6.72297468342721334061609536728, 7.20683429850699212519306839750, 8.488847613102835692075792235747
