| L(s) = 1 | + 209.·5-s + (775. + 471. i)7-s − 2.15e3i·11-s − 6.16e3i·13-s − 1.20e4·17-s − 3.52e4i·19-s − 3.86e4i·23-s − 3.43e4·25-s − 4.46e4i·29-s − 3.92e4i·31-s + (1.62e5 + 9.87e4i)35-s − 9.53e4·37-s + 3.59e5·41-s + 1.52e5·43-s − 5.22e5·47-s + ⋯ |
| L(s) = 1 | + 0.748·5-s + (0.854 + 0.519i)7-s − 0.488i·11-s − 0.777i·13-s − 0.593·17-s − 1.17i·19-s − 0.662i·23-s − 0.439·25-s − 0.339i·29-s − 0.236i·31-s + (0.639 + 0.389i)35-s − 0.309·37-s + 0.813·41-s + 0.293·43-s − 0.733·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0686 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0686 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.174794356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174794356\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-775. - 471. i)T \) |
| good | 5 | \( 1 - 209.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.15e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 6.16e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.52e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 3.86e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 4.46e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.92e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 9.53e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.22e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.02e4iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.58e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.42e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.42e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.71e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 1.96e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.72e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.57e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7iT - 8.07e13T^{2} \) |
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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
−10.70006900928762988767395541660, −9.519427887094501159912216808131, −8.676322131839851044882467621048, −7.74747705310515996231219353253, −6.39601069146260982961240307912, −5.50113461702600591900789798765, −4.52433737397656033266824045436, −2.86625560717962111477970079477, −1.88298851425727505901775127964, −0.48084195043951146869959708830,
1.35003009209943599500158213893, 2.12085954080265360350546634410, 3.84431258512652518287223809086, 4.87291493184342804744747979667, 5.98702831758723675089565251693, 7.09895680525900502325844177509, 8.063528588288816653191919092967, 9.213165143363381201516719724228, 10.05237767451851644141757043791, 10.98543896256466747889938905207
