| L(s) = 1 | − 20.9·2-s − 72.2·4-s + 3.57e3·7-s + 1.22e4·8-s + 7.45e4·11-s − 3.44e4·13-s − 7.49e4·14-s − 2.19e5·16-s − 3.72e5·17-s − 8.35e5·19-s − 1.56e6·22-s − 3.18e4·23-s + 7.23e5·26-s − 2.58e5·28-s + 6.73e6·29-s − 4.04e6·31-s − 1.66e6·32-s + 7.81e6·34-s − 6.29e6·37-s + 1.75e7·38-s − 1.10e7·41-s + 1.42e7·43-s − 5.38e6·44-s + 6.67e5·46-s + 2.76e7·47-s − 2.75e7·49-s + 2.49e6·52-s + ⋯ |
| L(s) = 1 | − 0.926·2-s − 0.141·4-s + 0.562·7-s + 1.05·8-s + 1.53·11-s − 0.334·13-s − 0.521·14-s − 0.838·16-s − 1.08·17-s − 1.47·19-s − 1.42·22-s − 0.0237·23-s + 0.310·26-s − 0.0794·28-s + 1.76·29-s − 0.786·31-s − 0.280·32-s + 1.00·34-s − 0.552·37-s + 1.36·38-s − 0.612·41-s + 0.635·43-s − 0.216·44-s + 0.0219·46-s + 0.826·47-s − 0.683·49-s + 0.0472·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 20.9T + 512T^{2} \) |
| 7 | \( 1 - 3.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.44e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.72e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.18e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.52e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.16e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.77e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.06e9T + 7.60e17T^{2} \) |
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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
−10.04965097718310947596081851223, −8.844957685802055434451279770318, −8.621562978731756288219487120609, −7.27102996688723351483244222904, −6.36981728224718263900661685008, −4.74136685261975047269765766676, −4.02182167026254755442774334219, −2.14665727510170189400593457644, −1.17381980457379103552844291362, 0,
1.17381980457379103552844291362, 2.14665727510170189400593457644, 4.02182167026254755442774334219, 4.74136685261975047269765766676, 6.36981728224718263900661685008, 7.27102996688723351483244222904, 8.621562978731756288219487120609, 8.844957685802055434451279770318, 10.04965097718310947596081851223
