| L(s) = 1 | − 4.86·2-s + 9·3-s − 8.29·4-s + 70.0·5-s − 43.8·6-s − 29.0·7-s + 196.·8-s + 81·9-s − 340.·10-s − 79.3·11-s − 74.6·12-s − 311.·13-s + 141.·14-s + 630.·15-s − 689.·16-s + 161.·17-s − 394.·18-s + 1.13e3·19-s − 580.·20-s − 261.·21-s + 386.·22-s − 3.91e3·23-s + 1.76e3·24-s + 1.77e3·25-s + 1.51e3·26-s + 729·27-s + 240.·28-s + ⋯ |
| L(s) = 1 | − 0.860·2-s + 0.577·3-s − 0.259·4-s + 1.25·5-s − 0.496·6-s − 0.223·7-s + 1.08·8-s + 0.333·9-s − 1.07·10-s − 0.197·11-s − 0.149·12-s − 0.510·13-s + 0.192·14-s + 0.723·15-s − 0.673·16-s + 0.135·17-s − 0.286·18-s + 0.718·19-s − 0.324·20-s − 0.129·21-s + 0.170·22-s − 1.54·23-s + 0.625·24-s + 0.569·25-s + 0.439·26-s + 0.192·27-s + 0.0580·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 157 | \( 1 - 2.46e4T \) |
| good | 2 | \( 1 + 4.86T + 32T^{2} \) |
| 5 | \( 1 - 70.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 29.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 79.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 311.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 161.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.44e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.50e4T + 8.58e9T^{2} \) |
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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
−9.617988814550884191795805014763, −9.198475038135192815616037464472, −8.044168007883015923996752185152, −7.36983862857223873683623257456, −6.05279464073822091387282693906, −5.09038123778317525604936411495, −3.79004659391334195431935100500, −2.33027042096206619034497771944, −1.47534864030260405243087101177, 0,
1.47534864030260405243087101177, 2.33027042096206619034497771944, 3.79004659391334195431935100500, 5.09038123778317525604936411495, 6.05279464073822091387282693906, 7.36983862857223873683623257456, 8.044168007883015923996752185152, 9.198475038135192815616037464472, 9.617988814550884191795805014763
