| L(s) = 1 | + 1.78·2-s + 3.10·3-s + 1.17·4-s + 5.53·6-s + 3.64·7-s − 1.47·8-s + 6.64·9-s − 5.44·11-s + 3.64·12-s − 3.49·13-s + 6.49·14-s − 4.96·16-s − 3.61·17-s + 11.8·18-s + 2.90·19-s + 11.3·21-s − 9.69·22-s − 7.81·23-s − 4.57·24-s − 6.22·26-s + 11.3·27-s + 4.27·28-s + 29-s + 3.40·31-s − 5.91·32-s − 16.8·33-s − 6.44·34-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 1.79·3-s + 0.586·4-s + 2.25·6-s + 1.37·7-s − 0.520·8-s + 2.21·9-s − 1.64·11-s + 1.05·12-s − 0.969·13-s + 1.73·14-s − 1.24·16-s − 0.876·17-s + 2.79·18-s + 0.665·19-s + 2.47·21-s − 2.06·22-s − 1.62·23-s − 0.933·24-s − 1.22·26-s + 2.17·27-s + 0.808·28-s + 0.185·29-s + 0.611·31-s − 1.04·32-s − 2.94·33-s − 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.711413877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.711413877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.44T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 7.44T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 - 2.41T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 0.901T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−10.30563615581845571851578403755, −9.479152765037758098767220612335, −8.385404129059849695531237236733, −7.954777693769985241687545538931, −7.11544730883696346092105848453, −5.52063277050490981102167749029, −4.68149609414940693366259110575, −4.00092429706720159523728519836, −2.65691115403362873782461279596, −2.23622422821556553393172854344,
2.23622422821556553393172854344, 2.65691115403362873782461279596, 4.00092429706720159523728519836, 4.68149609414940693366259110575, 5.52063277050490981102167749029, 7.11544730883696346092105848453, 7.954777693769985241687545538931, 8.385404129059849695531237236733, 9.479152765037758098767220612335, 10.30563615581845571851578403755
