L(s) = 1 | + 1.73·2-s − 2.73·3-s + 0.999·4-s − 4.73·6-s − 2·7-s − 1.73·8-s + 4.46·9-s − 4.73·11-s − 2.73·12-s − 13-s − 3.46·14-s − 5·16-s + 3.46·17-s + 7.73·18-s − 6.19·19-s + 5.46·21-s − 8.19·22-s − 1.26·23-s + 4.73·24-s − 1.73·26-s − 3.99·27-s − 1.99·28-s − 2.53·29-s + 10.1·31-s − 5.19·32-s + 12.9·33-s + 5.99·34-s + ⋯ |
L(s) = 1 | + 1.22·2-s − 1.57·3-s + 0.499·4-s − 1.93·6-s − 0.755·7-s − 0.612·8-s + 1.48·9-s − 1.42·11-s − 0.788·12-s − 0.277·13-s − 0.925·14-s − 1.25·16-s + 0.840·17-s + 1.82·18-s − 1.42·19-s + 1.19·21-s − 1.74·22-s − 0.264·23-s + 0.965·24-s − 0.339·26-s − 0.769·27-s − 0.377·28-s − 0.470·29-s + 1.83·31-s − 0.918·32-s + 2.25·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.38855442644562224996178361785, −10.49514763031885498746455965242, −9.700906042718965173474722317460, −8.043302671108735330197913120116, −6.62196574101091805992619073964, −5.99786740878866282735009424358, −5.16523178123401442521525889704, −4.33519134317467989512880406743, −2.83749279817631023161663458700, 0,
2.83749279817631023161663458700, 4.33519134317467989512880406743, 5.16523178123401442521525889704, 5.99786740878866282735009424358, 6.62196574101091805992619073964, 8.043302671108735330197913120116, 9.700906042718965173474722317460, 10.49514763031885498746455965242, 11.38855442644562224996178361785