| L(s) = 1 | + 2.18e3·3-s − 4.20e4·5-s − 3.85e5·7-s + 4.78e6·9-s − 2.04e7·11-s + 7.52e7·13-s − 9.18e7·15-s + 6.39e8·17-s − 2.67e8·19-s − 8.43e8·21-s − 5.71e8·23-s − 2.87e10·25-s + 1.04e10·27-s − 6.24e10·29-s + 1.24e9·31-s − 4.47e10·33-s + 1.62e10·35-s − 4.08e11·37-s + 1.64e11·39-s − 1.37e12·41-s + 1.92e12·43-s − 2.00e11·45-s + 4.74e12·47-s − 4.59e12·49-s + 1.39e12·51-s − 3.26e12·53-s + 8.58e11·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.240·5-s − 0.177·7-s + 1/3·9-s − 0.316·11-s + 0.332·13-s − 0.138·15-s + 0.378·17-s − 0.0685·19-s − 0.102·21-s − 0.0350·23-s − 0.942·25-s + 0.192·27-s − 0.672·29-s + 0.00811·31-s − 0.182·33-s + 0.0425·35-s − 0.706·37-s + 0.191·39-s − 1.10·41-s + 1.07·43-s − 0.0801·45-s + 1.36·47-s − 0.968·49-s + 0.218·51-s − 0.381·53-s + 0.0760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{7} T \) |
| good | 5 | \( 1 + 8402 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 7872 p^{2} T + p^{15} T^{2} \) |
| 11 | \( 1 + 20444300 T + p^{15} T^{2} \) |
| 13 | \( 1 - 5786254 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 639707746 T + p^{15} T^{2} \) |
| 19 | \( 1 + 267006356 T + p^{15} T^{2} \) |
| 23 | \( 1 + 571925576 T + p^{15} T^{2} \) |
| 29 | \( 1 + 62437913154 T + p^{15} T^{2} \) |
| 31 | \( 1 - 1243357000 T + p^{15} T^{2} \) |
| 37 | \( 1 + 408247831602 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1376815491990 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1923521494772 T + p^{15} T^{2} \) |
| 47 | \( 1 - 4743411679104 T + p^{15} T^{2} \) |
| 53 | \( 1 + 3260491936570 T + p^{15} T^{2} \) |
| 59 | \( 1 + 6734040423500 T + p^{15} T^{2} \) |
| 61 | \( 1 + 10498750519274 T + p^{15} T^{2} \) |
| 67 | \( 1 + 42186307399892 T + p^{15} T^{2} \) |
| 71 | \( 1 + 81526680652600 T + p^{15} T^{2} \) |
| 73 | \( 1 + 37550447155142 T + p^{15} T^{2} \) |
| 79 | \( 1 + 169301409686344 T + p^{15} T^{2} \) |
| 83 | \( 1 + 357833604131476 T + p^{15} T^{2} \) |
| 89 | \( 1 + 157672146871542 T + p^{15} T^{2} \) |
| 97 | \( 1 + 1220442820436926 T + p^{15} T^{2} \) |
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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
−11.96985352564320719570460868448, −10.63687054323431130162794050931, −9.460387790524266453440807166320, −8.284910904292455410508944414595, −7.22217764598194354043969403902, −5.72088711182041834446617955579, −4.15658736066153881205692465589, −2.99649769346770320620483084137, −1.59561888710347696838953795528, 0,
1.59561888710347696838953795528, 2.99649769346770320620483084137, 4.15658736066153881205692465589, 5.72088711182041834446617955579, 7.22217764598194354043969403902, 8.284910904292455410508944414595, 9.460387790524266453440807166320, 10.63687054323431130162794050931, 11.96985352564320719570460868448
