L(s) = 1 | + 2-s + 2.41·3-s − 4-s + 2.41·6-s + 2·7-s − 3·8-s + 2.82·9-s + 11-s − 2.41·12-s + 0.414·13-s + 2·14-s − 16-s + 4.82·17-s + 2.82·18-s + 8.65·19-s + 4.82·21-s + 22-s − 6.07·23-s − 7.24·24-s + 0.414·26-s − 0.414·27-s − 2·28-s − 2.17·29-s + 0.171·31-s + 5·32-s + 2.41·33-s + 4.82·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.39·3-s − 0.5·4-s + 0.985·6-s + 0.755·7-s − 1.06·8-s + 0.942·9-s + 0.301·11-s − 0.696·12-s + 0.114·13-s + 0.534·14-s − 0.250·16-s + 1.17·17-s + 0.666·18-s + 1.98·19-s + 1.05·21-s + 0.213·22-s − 1.26·23-s − 1.47·24-s + 0.0812·26-s − 0.0797·27-s − 0.377·28-s − 0.403·29-s + 0.0308·31-s + 0.883·32-s + 0.420·33-s + 0.828·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.374959149\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.374959149\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 151 | \( 1 - T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 8.65T + 19T^{2} \) |
| 23 | \( 1 + 6.07T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 0.171T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.536088210200579203089613447773, −7.72921946675074014166449792538, −7.43978566759270123967893290381, −5.98964668593628328842075824491, −5.42743412408699492573044985572, −4.55586950936724981202614410813, −3.68111807386587632135390846084, −3.28677898091630887899537334108, −2.26804313650135889069297927576, −1.10940532010842250096040164044,
1.10940532010842250096040164044, 2.26804313650135889069297927576, 3.28677898091630887899537334108, 3.68111807386587632135390846084, 4.55586950936724981202614410813, 5.42743412408699492573044985572, 5.98964668593628328842075824491, 7.43978566759270123967893290381, 7.72921946675074014166449792538, 8.536088210200579203089613447773