L(s) = 1 | − 8.77·2-s + 45.0·4-s + 17.5·5-s + 12.9·7-s − 114.·8-s − 153.·10-s + 434.·11-s − 637.·13-s − 113.·14-s − 433.·16-s + 850.·17-s + 1.82e3·19-s + 789.·20-s − 3.81e3·22-s − 715.·23-s − 2.81e3·25-s + 5.59e3·26-s + 584.·28-s − 1.13e3·29-s − 9.55e3·31-s + 7.48e3·32-s − 7.46e3·34-s + 226.·35-s − 107.·37-s − 1.59e4·38-s − 2.01e3·40-s + 6.61e3·41-s + ⋯ |
L(s) = 1 | − 1.55·2-s + 1.40·4-s + 0.313·5-s + 0.0999·7-s − 0.634·8-s − 0.486·10-s + 1.08·11-s − 1.04·13-s − 0.155·14-s − 0.423·16-s + 0.713·17-s + 1.15·19-s + 0.441·20-s − 1.67·22-s − 0.282·23-s − 0.901·25-s + 1.62·26-s + 0.140·28-s − 0.250·29-s − 1.78·31-s + 1.29·32-s − 1.10·34-s + 0.0313·35-s − 0.0128·37-s − 1.79·38-s − 0.198·40-s + 0.614·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 + 8.77T + 32T^{2} \) |
| 5 | \( 1 - 17.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 12.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 637.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 850.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 715.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 107.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.61e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 2.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.10e4T + 8.58e9T^{2} \) |
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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
−9.601871555479954582957066003300, −8.972779410942663568584912056508, −7.79499995985489294852380891859, −7.31549895753212724885721148821, −6.22509729197012738534641921178, −5.05055271920208815375771570505, −3.57638146121249004986794314941, −2.10374779654820739378185560689, −1.22735994179874913519841986374, 0,
1.22735994179874913519841986374, 2.10374779654820739378185560689, 3.57638146121249004986794314941, 5.05055271920208815375771570505, 6.22509729197012738534641921178, 7.31549895753212724885721148821, 7.79499995985489294852380891859, 8.972779410942663568584912056508, 9.601871555479954582957066003300