| L(s) = 1 | + 3·3-s + 2.58·5-s − 33.7·7-s + 9·9-s + 33.1·11-s + 4.33·13-s + 7.74·15-s + 11.1·17-s − 121.·19-s − 101.·21-s − 14.8·23-s − 118.·25-s + 27·27-s − 272.·29-s − 165.·31-s + 99.4·33-s − 87.1·35-s + 30.5·37-s + 12.9·39-s + 400.·41-s − 274.·43-s + 23.2·45-s − 487.·47-s + 795.·49-s + 33.4·51-s + 208.·53-s + 85.6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.231·5-s − 1.82·7-s + 0.333·9-s + 0.909·11-s + 0.0924·13-s + 0.133·15-s + 0.159·17-s − 1.47·19-s − 1.05·21-s − 0.134·23-s − 0.946·25-s + 0.192·27-s − 1.74·29-s − 0.956·31-s + 0.524·33-s − 0.421·35-s + 0.135·37-s + 0.0533·39-s + 1.52·41-s − 0.974·43-s + 0.0770·45-s − 1.51·47-s + 2.32·49-s + 0.0919·51-s + 0.540·53-s + 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 - 2.58T + 125T^{2} \) |
| 7 | \( 1 + 33.7T + 343T^{2} \) |
| 11 | \( 1 - 33.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.33T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 400.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 369.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 411.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 262.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 955.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−10.20017201587611867858319372315, −9.441106416873876225662774999360, −8.919575797686852375116026143187, −7.58126680421848766415273949221, −6.55370907152484727528657725749, −5.90425899453107667901883527662, −4.08717869489041217886104560001, −3.32903178247906982595508550717, −1.96176886693335888527386396680, 0,
1.96176886693335888527386396680, 3.32903178247906982595508550717, 4.08717869489041217886104560001, 5.90425899453107667901883527662, 6.55370907152484727528657725749, 7.58126680421848766415273949221, 8.919575797686852375116026143187, 9.441106416873876225662774999360, 10.20017201587611867858319372315
