| L(s) = 1 | + 6.54·3-s − 3.45·5-s − 7·7-s + 15.9·9-s − 27.2·11-s − 58.7·13-s − 22.5·15-s + 49.2·17-s − 157.·19-s − 45.8·21-s + 82.1·23-s − 113.·25-s − 72.7·27-s + 194.·29-s − 115.·31-s − 178.·33-s + 24.1·35-s − 327.·37-s − 384.·39-s − 136.·41-s + 311.·43-s − 54.8·45-s + 355.·47-s + 49·49-s + 322.·51-s − 677.·53-s + 94.1·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.308·5-s − 0.377·7-s + 0.588·9-s − 0.748·11-s − 1.25·13-s − 0.388·15-s + 0.703·17-s − 1.89·19-s − 0.476·21-s + 0.745·23-s − 0.904·25-s − 0.518·27-s + 1.24·29-s − 0.670·31-s − 0.943·33-s + 0.116·35-s − 1.45·37-s − 1.57·39-s − 0.518·41-s + 1.10·43-s − 0.181·45-s + 1.10·47-s + 0.142·49-s + 0.886·51-s − 1.75·53-s + 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 7T \) |
| good | 3 | \( 1 - 6.54T + 27T^{2} \) |
| 5 | \( 1 + 3.45T + 125T^{2} \) |
| 11 | \( 1 + 27.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 157.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 355.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 677.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 197.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 61.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 20.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 260.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 909.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 100.T + 9.12e5T^{2} \) |
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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
−10.08383274333008257855107407833, −9.252038593917326524057528073284, −8.348054755270989929869470409676, −7.70404391567219691941525168510, −6.73101194788399328209269071897, −5.31574399815440862225627537182, −4.10523801248342783445928634543, −2.99868283586374921286756551634, −2.14592274663672922733430785692, 0,
2.14592274663672922733430785692, 2.99868283586374921286756551634, 4.10523801248342783445928634543, 5.31574399815440862225627537182, 6.73101194788399328209269071897, 7.70404391567219691941525168510, 8.348054755270989929869470409676, 9.252038593917326524057528073284, 10.08383274333008257855107407833
