| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 4·13-s + 14-s + 16-s − 17-s − 6·19-s − 2·20-s − 25-s + 4·26-s + 28-s − 6·29-s − 8·31-s + 32-s − 34-s − 2·35-s − 2·37-s − 6·38-s − 2·40-s + 2·41-s + 10·43-s + 2·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.447·20-s − 1/5·25-s + 0.784·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s − 0.328·37-s − 0.973·38-s − 0.316·40-s + 0.312·41-s + 1.52·43-s + 0.291·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.92274730759740, −12.71327679519163, −12.17187814707726, −11.64717177801453, −11.19841472811385, −10.86735048447039, −10.64970038746289, −9.897867163629175, −9.165963383187680, −8.792674031507336, −8.409972478147540, −7.732778487120100, −7.421677603027563, −7.011898354941699, −6.210883818918686, −5.924817356490461, −5.495182627100394, −4.658365958421414, −4.329789769632085, −3.839409367685942, −3.506237084469219, −2.818767629552756, −1.982340706858156, −1.738474793808633, −0.7654306096275251, 0,
0.7654306096275251, 1.738474793808633, 1.982340706858156, 2.818767629552756, 3.506237084469219, 3.839409367685942, 4.329789769632085, 4.658365958421414, 5.495182627100394, 5.924817356490461, 6.210883818918686, 7.011898354941699, 7.421677603027563, 7.732778487120100, 8.409972478147540, 8.792674031507336, 9.165963383187680, 9.897867163629175, 10.64970038746289, 10.86735048447039, 11.19841472811385, 11.64717177801453, 12.17187814707726, 12.71327679519163, 12.92274730759740
