| L(s) = 1 | − 1.68·2-s − 5.17·4-s + 17.3·5-s + 7·7-s + 22.1·8-s − 29.1·10-s − 11·11-s − 70.1·13-s − 11.7·14-s + 4.17·16-s + 65.9·17-s − 105.·19-s − 89.8·20-s + 18.4·22-s − 96.1·23-s + 176.·25-s + 117.·26-s − 36.2·28-s − 196.·29-s − 25.2·31-s − 184.·32-s − 110.·34-s + 121.·35-s − 264.·37-s + 176.·38-s + 384.·40-s + 227.·41-s + ⋯ |
| L(s) = 1 | − 0.594·2-s − 0.646·4-s + 1.55·5-s + 0.377·7-s + 0.978·8-s − 0.923·10-s − 0.301·11-s − 1.49·13-s − 0.224·14-s + 0.0653·16-s + 0.941·17-s − 1.26·19-s − 1.00·20-s + 0.179·22-s − 0.871·23-s + 1.41·25-s + 0.889·26-s − 0.244·28-s − 1.25·29-s − 0.146·31-s − 1.01·32-s − 0.559·34-s + 0.587·35-s − 1.17·37-s + 0.753·38-s + 1.52·40-s + 0.867·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.68T + 8T^{2} \) |
| 5 | \( 1 - 17.3T + 125T^{2} \) |
| 13 | \( 1 + 70.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 25.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 459.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 6.15T + 1.03e5T^{2} \) |
| 53 | \( 1 + 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 334.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 93.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 39.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 494.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 390.T + 9.12e5T^{2} \) |
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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
−9.666207873847931507737312993973, −9.004955972354833013283745464261, −7.998300352545490045654547787918, −7.19145686426055099432910237219, −5.84291775896621342857185169605, −5.23449175601027695277963585688, −4.19645427945810496431276267893, −2.45198678787125339842704750283, −1.56903001544265939535233418110, 0,
1.56903001544265939535233418110, 2.45198678787125339842704750283, 4.19645427945810496431276267893, 5.23449175601027695277963585688, 5.84291775896621342857185169605, 7.19145686426055099432910237219, 7.998300352545490045654547787918, 9.004955972354833013283745464261, 9.666207873847931507737312993973
