L(s) = 1 | − 2.26e10·2-s + 3.67e20·4-s + 1.61e23·5-s + 2.93e28·7-s − 4.99e30·8-s − 3.65e33·10-s − 1.03e35·11-s − 9.30e35·13-s − 6.66e38·14-s + 5.90e40·16-s − 2.52e41·17-s − 4.10e41·19-s + 5.92e43·20-s + 2.36e45·22-s + 7.28e44·23-s − 4.17e46·25-s + 2.11e46·26-s + 1.07e49·28-s + 8.66e48·29-s + 5.42e49·31-s − 6.03e50·32-s + 5.72e51·34-s + 4.73e51·35-s + 2.44e52·37-s + 9.32e51·38-s − 8.04e53·40-s − 8.84e52·41-s + ⋯ |
L(s) = 1 | − 1.86·2-s + 2.49·4-s + 0.619·5-s + 1.43·7-s − 2.78·8-s − 1.15·10-s − 1.34·11-s − 0.0448·13-s − 2.68·14-s + 2.71·16-s − 1.51·17-s − 0.0596·19-s + 1.54·20-s + 2.52·22-s + 0.175·23-s − 0.616·25-s + 0.0837·26-s + 3.57·28-s + 0.886·29-s + 0.594·31-s − 2.28·32-s + 2.83·34-s + 0.889·35-s + 0.712·37-s + 0.111·38-s − 1.72·40-s − 0.0829·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(34)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{69}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + 2.26e10T + 1.47e20T^{2} \) |
| 5 | \( 1 - 1.61e23T + 6.77e46T^{2} \) |
| 7 | \( 1 - 2.93e28T + 4.18e56T^{2} \) |
| 11 | \( 1 + 1.03e35T + 5.93e69T^{2} \) |
| 13 | \( 1 + 9.30e35T + 4.30e74T^{2} \) |
| 17 | \( 1 + 2.52e41T + 2.75e82T^{2} \) |
| 19 | \( 1 + 4.10e41T + 4.74e85T^{2} \) |
| 23 | \( 1 - 7.28e44T + 1.72e91T^{2} \) |
| 29 | \( 1 - 8.66e48T + 9.56e97T^{2} \) |
| 31 | \( 1 - 5.42e49T + 8.34e99T^{2} \) |
| 37 | \( 1 - 2.44e52T + 1.17e105T^{2} \) |
| 41 | \( 1 + 8.84e52T + 1.13e108T^{2} \) |
| 43 | \( 1 + 3.35e54T + 2.76e109T^{2} \) |
| 47 | \( 1 - 1.34e56T + 1.07e112T^{2} \) |
| 53 | \( 1 - 3.87e57T + 3.36e115T^{2} \) |
| 59 | \( 1 - 2.62e59T + 4.43e118T^{2} \) |
| 61 | \( 1 + 4.77e59T + 4.14e119T^{2} \) |
| 67 | \( 1 + 2.36e61T + 2.22e122T^{2} \) |
| 71 | \( 1 + 2.36e61T + 1.08e124T^{2} \) |
| 73 | \( 1 - 7.86e61T + 6.96e124T^{2} \) |
| 79 | \( 1 + 2.74e63T + 1.38e127T^{2} \) |
| 83 | \( 1 + 9.16e63T + 3.78e128T^{2} \) |
| 89 | \( 1 - 3.36e65T + 4.06e130T^{2} \) |
| 97 | \( 1 - 6.50e66T + 1.29e133T^{2} \) |
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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
−10.14223790159457849409322887570, −8.874844805520706524159872466096, −8.149252440867280361816425317787, −7.26814248679887920449718494250, −6.00699427613460170744990652437, −4.73492720070418793023645048572, −2.54911642168043193285587331131, −2.04930748793261219294132245389, −1.04274823478568310987206575717, 0,
1.04274823478568310987206575717, 2.04930748793261219294132245389, 2.54911642168043193285587331131, 4.73492720070418793023645048572, 6.00699427613460170744990652437, 7.26814248679887920449718494250, 8.149252440867280361816425317787, 8.874844805520706524159872466096, 10.14223790159457849409322887570