| L(s) = 1 | − 7.82·3-s + 2.07·5-s + 34.1·9-s − 49.1·11-s + 44.8·13-s − 16.2·15-s − 26.5·17-s + 77.7·19-s − 55.7·23-s − 120.·25-s − 56.2·27-s + 121.·29-s + 305.·31-s + 384.·33-s + 77.1·37-s − 350.·39-s − 248.·41-s + 147.·43-s + 71.0·45-s + 269.·47-s + 207.·51-s − 141.·53-s − 102.·55-s − 608.·57-s − 424.·59-s + 587.·61-s + 93.1·65-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 0.185·5-s + 1.26·9-s − 1.34·11-s + 0.956·13-s − 0.279·15-s − 0.378·17-s + 0.938·19-s − 0.505·23-s − 0.965·25-s − 0.400·27-s + 0.777·29-s + 1.77·31-s + 2.02·33-s + 0.342·37-s − 1.44·39-s − 0.947·41-s + 0.521·43-s + 0.235·45-s + 0.837·47-s + 0.569·51-s − 0.365·53-s − 0.250·55-s − 1.41·57-s − 0.937·59-s + 1.23·61-s + 0.177·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 7.82T + 27T^{2} \) |
| 5 | \( 1 - 2.07T + 125T^{2} \) |
| 11 | \( 1 + 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 424.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 179.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 495.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−9.891425191535234459082446482468, −8.528389219356907136885906261727, −7.67347132308938388712881341662, −6.57921653727897942312079437815, −5.86649899573915052560316925046, −5.19449002473648229206522963413, −4.23856567075930751759195213981, −2.74615417524793132459023928891, −1.17802198936285649311244524486, 0,
1.17802198936285649311244524486, 2.74615417524793132459023928891, 4.23856567075930751759195213981, 5.19449002473648229206522963413, 5.86649899573915052560316925046, 6.57921653727897942312079437815, 7.67347132308938388712881341662, 8.528389219356907136885906261727, 9.891425191535234459082446482468
