| L(s) = 1 | + 2-s + 0.302·3-s + 4-s + 5-s + 0.302·6-s + 2.60·7-s + 8-s − 2.90·9-s + 10-s − 11-s + 0.302·12-s − 4.30·13-s + 2.60·14-s + 0.302·15-s + 16-s − 2.90·18-s + 0.697·19-s + 20-s + 0.788·21-s − 22-s − 2·23-s + 0.302·24-s + 25-s − 4.30·26-s − 1.78·27-s + 2.60·28-s − 6.60·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.174·3-s + 0.5·4-s + 0.447·5-s + 0.123·6-s + 0.984·7-s + 0.353·8-s − 0.969·9-s + 0.316·10-s − 0.301·11-s + 0.0874·12-s − 1.19·13-s + 0.696·14-s + 0.0781·15-s + 0.250·16-s − 0.685·18-s + 0.159·19-s + 0.223·20-s + 0.172·21-s − 0.213·22-s − 0.417·23-s + 0.0618·24-s + 0.200·25-s − 0.843·26-s − 0.344·27-s + 0.492·28-s − 1.22·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 0.697T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 0.697T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 + 0.302T + 67T^{2} \) |
| 71 | \( 1 - 0.697T + 71T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.42961081173554289547965515994, −6.83208779410768763410185290546, −5.78415653263534658968429374076, −5.36388161874023781501522931246, −4.86755909591906784264313266141, −3.91347976431103497914504207207, −3.08740026753318082394165663733, −2.24339346847094932228803445100, −1.69306437602520272580029042061, 0,
1.69306437602520272580029042061, 2.24339346847094932228803445100, 3.08740026753318082394165663733, 3.91347976431103497914504207207, 4.86755909591906784264313266141, 5.36388161874023781501522931246, 5.78415653263534658968429374076, 6.83208779410768763410185290546, 7.42961081173554289547965515994
