| L(s) = 1 | − 15.4·2-s − 207.·3-s − 274.·4-s + 1.54e3·5-s + 3.19e3·6-s + 3.95e3·7-s + 1.21e4·8-s + 2.32e4·9-s − 2.38e4·10-s − 2.26e4·11-s + 5.69e4·12-s − 6.02e4·13-s − 6.09e4·14-s − 3.20e5·15-s − 4.58e4·16-s + 5.54e3·17-s − 3.58e5·18-s + 8.32e5·19-s − 4.24e5·20-s − 8.20e5·21-s + 3.48e5·22-s − 1.66e6·23-s − 2.51e6·24-s + 4.36e5·25-s + 9.27e5·26-s − 7.47e5·27-s − 1.08e6·28-s + ⋯ |
| L(s) = 1 | − 0.680·2-s − 1.47·3-s − 0.536·4-s + 1.10·5-s + 1.00·6-s + 0.623·7-s + 1.04·8-s + 1.18·9-s − 0.752·10-s − 0.466·11-s + 0.793·12-s − 0.584·13-s − 0.424·14-s − 1.63·15-s − 0.175·16-s + 0.0160·17-s − 0.805·18-s + 1.46·19-s − 0.593·20-s − 0.920·21-s + 0.317·22-s − 1.24·23-s − 1.54·24-s + 0.223·25-s + 0.397·26-s − 0.270·27-s − 0.334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + 7.07e5T \) |
| good | 2 | \( 1 + 15.4T + 512T^{2} \) |
| 3 | \( 1 + 207.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.54e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.95e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.26e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.02e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.54e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.32e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.66e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 7.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.86e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.20e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.70e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.31e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.48e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.97e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.53e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.94e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.55e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.83e7T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27922405274446659403494904483, −13.09541719646556017312094165239, −11.63905639473787499893459927559, −10.38020777851385090902605512352, −9.516849270219041588863873823178, −7.64960335705068567976824629299, −5.80299210710396856808707981488, −4.87146126908411986537202873977, −1.52141380941163857753508989264, 0,
1.52141380941163857753508989264, 4.87146126908411986537202873977, 5.80299210710396856808707981488, 7.64960335705068567976824629299, 9.516849270219041588863873823178, 10.38020777851385090902605512352, 11.63905639473787499893459927559, 13.09541719646556017312094165239, 14.27922405274446659403494904483
