| L(s) = 1 | − 2.46·2-s + 4.05·4-s − 3.65·5-s − 5.05·8-s + 9.00·10-s − 0.406·11-s + 0.486·13-s + 4.32·16-s + 4.85·17-s + 1.97·19-s − 14.8·20-s + 22-s − 4.64·23-s + 8.38·25-s − 1.19·26-s − 7.64·29-s − 7.02·31-s − 0.539·32-s − 11.9·34-s + 2.32·37-s − 4.85·38-s + 18.4·40-s + 7.51·41-s − 2.32·43-s − 1.64·44-s + 11.4·46-s − 6.31·47-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.02·4-s − 1.63·5-s − 1.78·8-s + 2.84·10-s − 0.122·11-s + 0.135·13-s + 1.08·16-s + 1.17·17-s + 0.452·19-s − 3.31·20-s + 0.213·22-s − 0.969·23-s + 1.67·25-s − 0.234·26-s − 1.42·29-s − 1.26·31-s − 0.0953·32-s − 2.04·34-s + 0.382·37-s − 0.787·38-s + 2.92·40-s + 1.17·41-s − 0.354·43-s − 0.248·44-s + 1.68·46-s − 0.921·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 11 | \( 1 + 0.406T + 11T^{2} \) |
| 13 | \( 1 - 0.486T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 1.97T + 19T^{2} \) |
| 23 | \( 1 + 4.64T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 - 7.51T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−7.961817956632890341987015736621, −7.66156847796549699543078715972, −7.18795761745717522395745690486, −6.17982157665850064772503351367, −5.20547226858869477871649771379, −3.94109110643597913273445801217, −3.36998975564205608437255943129, −2.15190795374646663747122344116, −0.975693144638551746428826892199, 0,
0.975693144638551746428826892199, 2.15190795374646663747122344116, 3.36998975564205608437255943129, 3.94109110643597913273445801217, 5.20547226858869477871649771379, 6.17982157665850064772503351367, 7.18795761745717522395745690486, 7.66156847796549699543078715972, 7.961817956632890341987015736621
