L(s) = 1 | + 0.655·2-s − 1.57·4-s − 5-s − 0.777·7-s − 2.33·8-s − 0.655·10-s − 4.06·11-s − 13-s − 0.509·14-s + 1.60·16-s + 5.29·17-s + 6.65·19-s + 1.57·20-s − 2.66·22-s + 2.30·23-s + 25-s − 0.655·26-s + 1.22·28-s + 6.38·29-s − 8.94·31-s + 5.73·32-s + 3.46·34-s + 0.777·35-s − 2.80·37-s + 4.35·38-s + 2.33·40-s + 7.60·41-s + ⋯ |
L(s) = 1 | + 0.463·2-s − 0.785·4-s − 0.447·5-s − 0.293·7-s − 0.827·8-s − 0.207·10-s − 1.22·11-s − 0.277·13-s − 0.136·14-s + 0.402·16-s + 1.28·17-s + 1.52·19-s + 0.351·20-s − 0.568·22-s + 0.480·23-s + 0.200·25-s − 0.128·26-s + 0.230·28-s + 1.18·29-s − 1.60·31-s + 1.01·32-s + 0.594·34-s + 0.131·35-s − 0.461·37-s + 0.706·38-s + 0.369·40-s + 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.655T + 2T^{2} \) |
| 7 | \( 1 + 0.777T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 - 2.30T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 0.307T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
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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
−7.69525015427714863500188359113, −7.40593659118047015113484789244, −6.19206196991138590711723231538, −5.37880783636124300996622726460, −5.07216932052585768750210591241, −4.14096172707395788641469974978, −3.21205417076425851155362046226, −2.86225715158227223845418742106, −1.15735604336170687004414838404, 0,
1.15735604336170687004414838404, 2.86225715158227223845418742106, 3.21205417076425851155362046226, 4.14096172707395788641469974978, 5.07216932052585768750210591241, 5.37880783636124300996622726460, 6.19206196991138590711723231538, 7.40593659118047015113484789244, 7.69525015427714863500188359113