| L(s) = 1 | + 8.51·3-s − 8.47·5-s + 45.4·9-s − 25.0·11-s + 21.5·13-s − 72.1·15-s − 101.·17-s − 145.·19-s − 177.·23-s − 53.1·25-s + 157.·27-s + 89.5·29-s + 211.·31-s − 212.·33-s − 272.·37-s + 183.·39-s + 50.4·41-s + 332.·43-s − 385.·45-s + 313.·47-s − 865.·51-s − 105.·53-s + 212.·55-s − 1.24e3·57-s − 465.·59-s + 32.8·61-s − 183.·65-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.758·5-s + 1.68·9-s − 0.685·11-s + 0.460·13-s − 1.24·15-s − 1.45·17-s − 1.76·19-s − 1.61·23-s − 0.425·25-s + 1.11·27-s + 0.573·29-s + 1.22·31-s − 1.12·33-s − 1.20·37-s + 0.754·39-s + 0.192·41-s + 1.17·43-s − 1.27·45-s + 0.973·47-s − 2.37·51-s − 0.273·53-s + 0.519·55-s − 2.88·57-s − 1.02·59-s + 0.0689·61-s − 0.349·65-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 | \( 1 - 8.51T + 27T^{2} \) |
| 5 | \( 1 + 8.47T + 125T^{2} \) |
| 11 | \( 1 + 25.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 465.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 32.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 55.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 19.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 167.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 538.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 597.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 755.T + 9.12e5T^{2} \) |
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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
−9.228716578991783828394216984407, −8.313395380168409099402166032270, −8.214341270934555687317765380350, −7.13084845236027296140616106234, −6.13291796721341881860108603602, −4.38132725023411831472728486495, −3.98640714771460314339528476005, −2.72778480519427438502592483238, −1.96793755408728214516250445062, 0,
1.96793755408728214516250445062, 2.72778480519427438502592483238, 3.98640714771460314339528476005, 4.38132725023411831472728486495, 6.13291796721341881860108603602, 7.13084845236027296140616106234, 8.214341270934555687317765380350, 8.313395380168409099402166032270, 9.228716578991783828394216984407
