| L(s) = 1 | − 2.48·2-s − 0.945·3-s + 4.15·4-s + 3.33·5-s + 2.34·6-s − 4.81·7-s − 5.35·8-s − 2.10·9-s − 8.27·10-s + 3.69·11-s − 3.93·12-s + 1.66·13-s + 11.9·14-s − 3.15·15-s + 4.96·16-s + 17-s + 5.22·18-s + 5.20·19-s + 13.8·20-s + 4.55·21-s − 9.16·22-s − 8.43·23-s + 5.06·24-s + 6.12·25-s − 4.13·26-s + 4.82·27-s − 20.0·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.546·3-s + 2.07·4-s + 1.49·5-s + 0.958·6-s − 1.81·7-s − 1.89·8-s − 0.701·9-s − 2.61·10-s + 1.11·11-s − 1.13·12-s + 0.461·13-s + 3.19·14-s − 0.814·15-s + 1.24·16-s + 0.242·17-s + 1.23·18-s + 1.19·19-s + 3.10·20-s + 0.993·21-s − 1.95·22-s − 1.75·23-s + 1.03·24-s + 1.22·25-s − 0.810·26-s + 0.929·27-s − 3.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5645931132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5645931132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 + 0.945T + 3T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 + 8.43T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 4.22T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 0.764T + 41T^{2} \) |
| 47 | \( 1 - 5.74T + 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 0.413T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 0.0889T + 89T^{2} \) |
| 97 | \( 1 + 0.0263T + 97T^{2} \) |
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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
−10.03923362248581002799634201243, −9.430712772926627130923508547747, −9.227617111182502921696247956693, −7.999283425231754231189648362395, −6.65795558885909061894874800277, −6.30957074986908724790466488814, −5.67098400939372872083937258127, −3.40301869810043284867247108514, −2.22431957232485353389539490034, −0.829129182955907352487188175723,
0.829129182955907352487188175723, 2.22431957232485353389539490034, 3.40301869810043284867247108514, 5.67098400939372872083937258127, 6.30957074986908724790466488814, 6.65795558885909061894874800277, 7.999283425231754231189648362395, 9.227617111182502921696247956693, 9.430712772926627130923508547747, 10.03923362248581002799634201243
