| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·7-s − 2·10-s + 3·11-s − 4·13-s + 4·14-s − 4·16-s + 2·17-s − 2·19-s + 2·20-s − 6·22-s − 5·23-s + 25-s + 8·26-s − 4·28-s + 29-s + 2·31-s + 8·32-s − 4·34-s − 2·35-s − 5·37-s + 4·38-s + 41-s − 43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s − 0.632·10-s + 0.904·11-s − 1.10·13-s + 1.06·14-s − 16-s + 0.485·17-s − 0.458·19-s + 0.447·20-s − 1.27·22-s − 1.04·23-s + 1/5·25-s + 1.56·26-s − 0.755·28-s + 0.185·29-s + 0.359·31-s + 1.41·32-s − 0.685·34-s − 0.338·35-s − 0.821·37-s + 0.648·38-s + 0.156·41-s − 0.152·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.461723857713474801357389225947, −8.554429216966450714847569756395, −7.81196145753633841412735874399, −6.85479276139241650623705644175, −6.34249079317202533056147024876, −5.09270572731924838661238404372, −3.92028000621048552286619245454, −2.58437848106042068431489338098, −1.49518596923197366998762084176, 0,
1.49518596923197366998762084176, 2.58437848106042068431489338098, 3.92028000621048552286619245454, 5.09270572731924838661238404372, 6.34249079317202533056147024876, 6.85479276139241650623705644175, 7.81196145753633841412735874399, 8.554429216966450714847569756395, 9.461723857713474801357389225947
