| L(s) = 1 | + 2·5-s + 11-s + 13-s + 17-s − 8·19-s − 4·23-s − 25-s − 6·29-s + 4·31-s − 6·37-s + 6·41-s − 12·43-s + 8·47-s − 7·49-s − 2·53-s + 2·55-s + 4·59-s − 2·61-s + 2·65-s − 12·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s + 2·85-s + 6·89-s − 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.274·53-s + 0.269·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182242991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182242991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.62816872470322, −12.05941415838568, −11.72081104608930, −11.11287188258996, −10.71189953739061, −10.15027510392374, −9.975838609292246, −9.388384408128609, −8.831763613703999, −8.578216563880243, −8.001630880156807, −7.495008670027899, −6.911122922267065, −6.368681939719719, −6.046051572661594, −5.716159795171014, −4.997817282407369, −4.549391032000459, −3.913395506326532, −3.578476902671605, −2.785514739191851, −2.203311313612255, −1.773066679463805, −1.310416445711714, −0.2645749514608882,
0.2645749514608882, 1.310416445711714, 1.773066679463805, 2.203311313612255, 2.785514739191851, 3.578476902671605, 3.913395506326532, 4.549391032000459, 4.997817282407369, 5.716159795171014, 6.046051572661594, 6.368681939719719, 6.911122922267065, 7.495008670027899, 8.001630880156807, 8.578216563880243, 8.831763613703999, 9.388384408128609, 9.975838609292246, 10.15027510392374, 10.71189953739061, 11.11287188258996, 11.72081104608930, 12.05941415838568, 12.62816872470322
