| L(s) = 1 | + 20.3·2-s + 287.·4-s − 405.·5-s + 52.5·7-s + 3.24e3·8-s − 8.25e3·10-s − 2.54e3·11-s + 7.13e3·13-s + 1.07e3·14-s + 2.93e4·16-s + 2.25e4·17-s − 6.39e3·19-s − 1.16e5·20-s − 5.18e4·22-s − 9.98e4·23-s + 8.59e4·25-s + 1.45e5·26-s + 1.51e4·28-s − 5.60e3·29-s − 1.63e5·31-s + 1.83e5·32-s + 4.58e5·34-s − 2.13e4·35-s + 1.98e5·37-s − 1.30e5·38-s − 1.31e6·40-s − 7.42e5·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 1.44·5-s + 0.0579·7-s + 2.24·8-s − 2.61·10-s − 0.576·11-s + 0.901·13-s + 0.104·14-s + 1.79·16-s + 1.11·17-s − 0.213·19-s − 3.25·20-s − 1.03·22-s − 1.71·23-s + 1.10·25-s + 1.62·26-s + 0.130·28-s − 0.0427·29-s − 0.983·31-s + 0.987·32-s + 2.00·34-s − 0.0839·35-s + 0.643·37-s − 0.385·38-s − 3.24·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 - 20.3T + 128T^{2} \) |
| 5 | \( 1 + 405.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 52.5T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.13e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.25e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.39e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.60e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.63e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.98e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.42e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 7.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.08e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.07e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.55e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.83e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.94e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.91e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.05e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.18e7T + 8.07e13T^{2} \) |
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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
−10.07981361862936816065929508799, −8.246230096744988742415763370547, −7.69183966980252370143537388559, −6.58006429626823545390048330873, −5.62586420181178725252706301619, −4.64063947216201107373650154599, −3.72521528130531241163976466955, −3.21184038470250593399170772530, −1.71896771378764195089523969720, 0,
1.71896771378764195089523969720, 3.21184038470250593399170772530, 3.72521528130531241163976466955, 4.64063947216201107373650154599, 5.62586420181178725252706301619, 6.58006429626823545390048330873, 7.69183966980252370143537388559, 8.246230096744988742415763370547, 10.07981361862936816065929508799
