| L(s) = 1 | − 3.57·3-s − 13.4·5-s − 14.1·9-s − 0.813·11-s + 34.9·13-s + 48.2·15-s + 117.·17-s + 93.2·19-s − 120.·23-s + 56.6·25-s + 147.·27-s + 8.56·29-s + 82.1·31-s + 2.91·33-s + 28.8·37-s − 125.·39-s − 70.5·41-s − 417.·43-s + 191.·45-s − 338.·47-s − 421.·51-s + 149.·53-s + 10.9·55-s − 333.·57-s + 94.1·59-s − 120.·61-s − 471.·65-s + ⋯ |
| L(s) = 1 | − 0.688·3-s − 1.20·5-s − 0.525·9-s − 0.0222·11-s + 0.745·13-s + 0.830·15-s + 1.67·17-s + 1.12·19-s − 1.09·23-s + 0.453·25-s + 1.05·27-s + 0.0548·29-s + 0.475·31-s + 0.0153·33-s + 0.128·37-s − 0.513·39-s − 0.268·41-s − 1.47·43-s + 0.633·45-s − 1.04·47-s − 1.15·51-s + 0.386·53-s + 0.0268·55-s − 0.775·57-s + 0.207·59-s − 0.252·61-s − 0.899·65-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 | \( 1 + 3.57T + 27T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 11 | \( 1 + 0.813T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.56T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 94.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 792.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 550.T + 9.12e5T^{2} \) |
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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
−9.612955936633358119035786092756, −8.262616850669063984572345118750, −7.969413773387493681564676716367, −6.84531822817553192067199005928, −5.83478179653561167573232596452, −5.09342385535846629269960596578, −3.84716770434027941673646726498, −3.10483405968378178166773960212, −1.16799725564934247889342696382, 0,
1.16799725564934247889342696382, 3.10483405968378178166773960212, 3.84716770434027941673646726498, 5.09342385535846629269960596578, 5.83478179653561167573232596452, 6.84531822817553192067199005928, 7.969413773387493681564676716367, 8.262616850669063984572345118750, 9.612955936633358119035786092756
