| L(s) = 1 | + 14.6·2-s + 27·3-s + 85.2·4-s + 493.·5-s + 394.·6-s − 368.·7-s − 624.·8-s + 729·9-s + 7.21e3·10-s + 7.53e3·11-s + 2.30e3·12-s − 417.·13-s − 5.37e3·14-s + 1.33e4·15-s − 2.00e4·16-s − 2.48e4·17-s + 1.06e4·18-s + 2.08e4·19-s + 4.20e4·20-s − 9.94e3·21-s + 1.10e5·22-s + 1.21e4·23-s − 1.68e4·24-s + 1.65e5·25-s − 6.09e3·26-s + 1.96e4·27-s − 3.13e4·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.577·3-s + 0.665·4-s + 1.76·5-s + 0.745·6-s − 0.405·7-s − 0.431·8-s + 0.333·9-s + 2.28·10-s + 1.70·11-s + 0.384·12-s − 0.0526·13-s − 0.523·14-s + 1.01·15-s − 1.22·16-s − 1.22·17-s + 0.430·18-s + 0.697·19-s + 1.17·20-s − 0.234·21-s + 2.20·22-s + 0.208·23-s − 0.249·24-s + 2.12·25-s − 0.0679·26-s + 0.192·27-s − 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.635675466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.635675466\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 14.6T + 128T^{2} \) |
| 5 | \( 1 - 493.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 368.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.53e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 417.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.08e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 5.05e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.88e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.36e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.13e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.59e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.64e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.90e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.89e5T + 8.07e13T^{2} \) |
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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
−13.35542134582203225588260972814, −12.79485255786920480154255640951, −11.31185546317560507606108553492, −9.511676519265972671186733211615, −9.154600961038637124890997688877, −6.69696087036085710803563500628, −5.95719082077408858737714045532, −4.50485095219149702262254865755, −3.06209934128242571816145236537, −1.71519289393176200647452926486,
1.71519289393176200647452926486, 3.06209934128242571816145236537, 4.50485095219149702262254865755, 5.95719082077408858737714045532, 6.69696087036085710803563500628, 9.154600961038637124890997688877, 9.511676519265972671186733211615, 11.31185546317560507606108553492, 12.79485255786920480154255640951, 13.35542134582203225588260972814
