| L(s) = 1 | − 0.691·2-s − 3·3-s − 7.52·4-s + 5·5-s + 2.07·6-s − 14.5·7-s + 10.7·8-s + 9·9-s − 3.45·10-s + 36.8·11-s + 22.5·12-s − 23.9·13-s + 10.0·14-s − 15·15-s + 52.7·16-s + 62.8·17-s − 6.22·18-s − 94.1·19-s − 37.6·20-s + 43.7·21-s − 25.4·22-s + 185.·23-s − 32.2·24-s + 25·25-s + 16.5·26-s − 27·27-s + 109.·28-s + ⋯ |
| L(s) = 1 | − 0.244·2-s − 0.577·3-s − 0.940·4-s + 0.447·5-s + 0.141·6-s − 0.786·7-s + 0.474·8-s + 0.333·9-s − 0.109·10-s + 1.00·11-s + 0.542·12-s − 0.510·13-s + 0.192·14-s − 0.258·15-s + 0.824·16-s + 0.897·17-s − 0.0815·18-s − 1.13·19-s − 0.420·20-s + 0.454·21-s − 0.246·22-s + 1.68·23-s − 0.273·24-s + 0.200·25-s + 0.124·26-s − 0.192·27-s + 0.739·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 0.691T + 8T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 - 36.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 11.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 61.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 636.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 379.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 623.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 298.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 524.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 447.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 558.T + 9.12e5T^{2} \) |
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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
−10.15867060504802352030157116514, −9.363192040235874905048343646027, −8.793178267373351469376739021283, −7.39441827492440061712253587055, −6.45628805481239957303103005381, −5.46920412329107535706262119468, −4.47359411889405413605184688862, −3.30012771818319658793766011066, −1.36958129817956984038366708115, 0,
1.36958129817956984038366708115, 3.30012771818319658793766011066, 4.47359411889405413605184688862, 5.46920412329107535706262119468, 6.45628805481239957303103005381, 7.39441827492440061712253587055, 8.793178267373351469376739021283, 9.363192040235874905048343646027, 10.15867060504802352030157116514
