| L(s) = 1 | + 3.77·2-s + 6.27·4-s + 5.74·5-s − 24.4·7-s − 6.51·8-s + 21.6·10-s − 9.31·11-s + 17.0·13-s − 92.2·14-s − 74.8·16-s − 19.6·17-s − 15.1·19-s + 36.0·20-s − 35.2·22-s − 108.·23-s − 92.0·25-s + 64.5·26-s − 153.·28-s − 237.·29-s + 184.·31-s − 230.·32-s − 74.3·34-s − 140.·35-s + 155.·37-s − 57.1·38-s − 37.3·40-s − 261.·41-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.784·4-s + 0.513·5-s − 1.31·7-s − 0.287·8-s + 0.686·10-s − 0.255·11-s + 0.364·13-s − 1.76·14-s − 1.16·16-s − 0.280·17-s − 0.182·19-s + 0.403·20-s − 0.341·22-s − 0.983·23-s − 0.736·25-s + 0.486·26-s − 1.03·28-s − 1.51·29-s + 1.06·31-s − 1.27·32-s − 0.375·34-s − 0.677·35-s + 0.690·37-s − 0.244·38-s − 0.147·40-s − 0.997·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 - 3.77T + 8T^{2} \) |
| 5 | \( 1 - 5.74T + 125T^{2} \) |
| 7 | \( 1 + 24.4T + 343T^{2} \) |
| 11 | \( 1 + 9.31T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 555.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 220.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 436.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 924.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 937.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 13.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 26.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 136.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 360.T + 9.12e5T^{2} \) |
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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
−9.895055937201679925740759211335, −9.385483113069572612789835614265, −8.127881331358214932996761035180, −6.70914659967265487213575640638, −6.14793518531492457189565926465, −5.35515632370156138259855115701, −4.11378452272330592755727104254, −3.30176141312288148385798612227, −2.18892084106219494916645563457, 0,
2.18892084106219494916645563457, 3.30176141312288148385798612227, 4.11378452272330592755727104254, 5.35515632370156138259855115701, 6.14793518531492457189565926465, 6.70914659967265487213575640638, 8.127881331358214932996761035180, 9.385483113069572612789835614265, 9.895055937201679925740759211335
