L(s) = 1 | − 11.0·2-s + 9·3-s + 89.3·4-s − 99.1·6-s − 62.7·7-s − 631.·8-s + 81·9-s + 121·11-s + 803.·12-s − 986.·13-s + 691.·14-s + 4.09e3·16-s − 577.·17-s − 892.·18-s − 234.·19-s − 564.·21-s − 1.33e3·22-s + 2.02e3·23-s − 5.68e3·24-s + 1.08e4·26-s + 729·27-s − 5.60e3·28-s − 6.87e3·29-s − 689.·31-s − 2.49e4·32-s + 1.08e3·33-s + 6.36e3·34-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.79·4-s − 1.12·6-s − 0.484·7-s − 3.48·8-s + 0.333·9-s + 0.301·11-s + 1.61·12-s − 1.61·13-s + 0.942·14-s + 4.00·16-s − 0.485·17-s − 0.649·18-s − 0.149·19-s − 0.279·21-s − 0.587·22-s + 0.797·23-s − 2.01·24-s + 3.15·26-s + 0.192·27-s − 1.35·28-s − 1.51·29-s − 0.128·31-s − 4.30·32-s + 0.174·33-s + 0.944·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5456385770\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5456385770\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 11.0T + 32T^{2} \) |
| 7 | \( 1 + 62.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 986.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 577.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 234.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 689.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.32e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e4T + 8.58e9T^{2} \) |
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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
−9.321041668824211125828029794654, −8.877408030104398203449921882114, −7.906887674323616074999975530907, −7.12875780789010797304832420303, −6.67906231485508317901426492835, −5.29825501642823730828724990762, −3.53738465200249216471639324689, −2.50559906696440746850094637465, −1.76131292021094790208671385078, −0.41852828881075049549548697306,
0.41852828881075049549548697306, 1.76131292021094790208671385078, 2.50559906696440746850094637465, 3.53738465200249216471639324689, 5.29825501642823730828724990762, 6.67906231485508317901426492835, 7.12875780789010797304832420303, 7.906887674323616074999975530907, 8.877408030104398203449921882114, 9.321041668824211125828029794654