| L(s) = 1 | + 4.39·2-s − 27·3-s − 108.·4-s − 380.·5-s − 118.·6-s − 1.69e3·7-s − 1.04e3·8-s + 729·9-s − 1.67e3·10-s + 8.25e3·11-s + 2.93e3·12-s − 2.53e3·13-s − 7.45e3·14-s + 1.02e4·15-s + 9.34e3·16-s − 7.20e3·17-s + 3.20e3·18-s − 2.41e4·19-s + 4.13e4·20-s + 4.57e4·21-s + 3.62e4·22-s − 1.21e4·23-s + 2.80e4·24-s + 6.64e4·25-s − 1.11e4·26-s − 1.96e4·27-s + 1.84e5·28-s + ⋯ |
| L(s) = 1 | + 0.388·2-s − 0.577·3-s − 0.849·4-s − 1.36·5-s − 0.224·6-s − 1.86·7-s − 0.718·8-s + 0.333·9-s − 0.528·10-s + 1.87·11-s + 0.490·12-s − 0.319·13-s − 0.725·14-s + 0.785·15-s + 0.570·16-s − 0.355·17-s + 0.129·18-s − 0.807·19-s + 1.15·20-s + 1.07·21-s + 0.726·22-s − 0.208·23-s + 0.414·24-s + 0.850·25-s − 0.124·26-s − 0.192·27-s + 1.58·28-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
| \(L(4)\) |
\(\approx\) |
\(0.4226343166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4226343166\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 4.39T + 128T^{2} \) |
| 5 | \( 1 + 380.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.69e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.53e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.20e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.41e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.22e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.09e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.45e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.87e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.58e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.07e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.80e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.92e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.13e6T + 8.07e13T^{2} \) |
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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
−12.92584680749546218307591625369, −12.39263659318763466422051401287, −11.35889230122546302453295622907, −9.717849557498487293891309817961, −8.873255986283883470145599412142, −7.07575416694199762332971468028, −6.04146799358190666259957178634, −4.16916985757515553133286311332, −3.59752161211800683711514340130, −0.41364203385135991932112890257,
0.41364203385135991932112890257, 3.59752161211800683711514340130, 4.16916985757515553133286311332, 6.04146799358190666259957178634, 7.07575416694199762332971468028, 8.873255986283883470145599412142, 9.717849557498487293891309817961, 11.35889230122546302453295622907, 12.39263659318763466422051401287, 12.92584680749546218307591625369
