| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 4·13-s − 2·14-s + 16-s − 6·17-s − 7·19-s + 4·26-s − 2·28-s + 6·29-s − 10·31-s + 32-s − 6·34-s − 2·37-s − 7·38-s − 9·41-s + 43-s + 6·47-s − 3·49-s + 4·52-s − 12·53-s − 2·56-s + 6·58-s + 9·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 1.60·19-s + 0.784·26-s − 0.377·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 1.13·38-s − 1.40·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.554·52-s − 1.64·53-s − 0.267·56-s + 0.787·58-s + 1.17·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.190569814901606549284727733969, −6.92686504235386448612955826051, −6.59403437377756029928284888118, −5.95947500014190205746646080199, −5.01745812369187468281424687266, −4.14497966792055238316516491333, −3.58521693107662408024268631245, −2.58061102642640590420464631190, −1.67438667985480429158537768244, 0,
1.67438667985480429158537768244, 2.58061102642640590420464631190, 3.58521693107662408024268631245, 4.14497966792055238316516491333, 5.01745812369187468281424687266, 5.95947500014190205746646080199, 6.59403437377756029928284888118, 6.92686504235386448612955826051, 8.190569814901606549284727733969
