| L(s) = 1 | − 2.64·2-s − 1.03·3-s − 1.01·4-s + 2.72·6-s + 6.51·7-s + 23.8·8-s − 25.9·9-s + 11·11-s + 1.05·12-s − 33.1·13-s − 17.2·14-s − 54.8·16-s − 5.13·17-s + 68.5·18-s + 14.2·19-s − 6.71·21-s − 29.0·22-s + 121.·23-s − 24.5·24-s + 87.6·26-s + 54.6·27-s − 6.63·28-s + 41.4·29-s + 9.40·31-s − 45.8·32-s − 11.3·33-s + 13.5·34-s + ⋯ |
| L(s) = 1 | − 0.934·2-s − 0.198·3-s − 0.127·4-s + 0.185·6-s + 0.351·7-s + 1.05·8-s − 0.960·9-s + 0.301·11-s + 0.0253·12-s − 0.707·13-s − 0.328·14-s − 0.856·16-s − 0.0732·17-s + 0.897·18-s + 0.171·19-s − 0.0698·21-s − 0.281·22-s + 1.09·23-s − 0.209·24-s + 0.661·26-s + 0.389·27-s − 0.0448·28-s + 0.265·29-s + 0.0545·31-s − 0.253·32-s − 0.0598·33-s + 0.0684·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 2.64T + 8T^{2} \) |
| 3 | \( 1 + 1.03T + 27T^{2} \) |
| 7 | \( 1 - 6.51T + 343T^{2} \) |
| 13 | \( 1 + 33.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.13T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.40T + 2.97e4T^{2} \) |
| 37 | \( 1 + 314.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 166.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 177.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 668.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 739.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 44.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 211.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 463.T + 9.12e5T^{2} \) |
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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
−8.792404946863142534512177240115, −8.230653302388245688577864545497, −7.36218301085996076287602632216, −6.56021166496657679241629324431, −5.30899930210764815481619345174, −4.78209871180562125010862180035, −3.51433567764545911270235424881, −2.27710362339507527639804956143, −1.04847951075936641570376153344, 0,
1.04847951075936641570376153344, 2.27710362339507527639804956143, 3.51433567764545911270235424881, 4.78209871180562125010862180035, 5.30899930210764815481619345174, 6.56021166496657679241629324431, 7.36218301085996076287602632216, 8.230653302388245688577864545497, 8.792404946863142534512177240115
