| L(s) = 1 | + 12·2-s − 8.04e3·4-s + 3.02e4·5-s + 2.35e5·7-s − 1.94e5·8-s + 3.62e5·10-s + 1.11e7·11-s + 8.04e6·13-s + 2.82e6·14-s + 6.35e7·16-s + 1.17e8·17-s − 2.14e8·19-s − 2.43e8·20-s + 1.34e8·22-s − 8.30e8·23-s − 3.08e8·25-s + 9.65e7·26-s − 1.89e9·28-s + 1.25e9·29-s + 6.15e9·31-s + 2.35e9·32-s + 1.40e9·34-s + 7.10e9·35-s − 5.49e9·37-s − 2.56e9·38-s − 5.88e9·40-s + 4.67e9·41-s + ⋯ |
| L(s) = 1 | + 0.132·2-s − 0.982·4-s + 0.864·5-s + 0.755·7-s − 0.262·8-s + 0.114·10-s + 1.90·11-s + 0.462·13-s + 0.100·14-s + 0.947·16-s + 1.18·17-s − 1.04·19-s − 0.849·20-s + 0.252·22-s − 1.16·23-s − 0.252·25-s + 0.0613·26-s − 0.741·28-s + 0.390·29-s + 1.24·31-s + 0.388·32-s + 0.156·34-s + 0.653·35-s − 0.352·37-s − 0.138·38-s − 0.227·40-s + 0.153·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.971306818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.971306818\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3 p^{2} T + p^{13} T^{2} \) |
| 5 | \( 1 - 6042 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 33584 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 1016628 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 8049614 T + p^{13} T^{2} \) |
| 17 | \( 1 - 117494622 T + p^{13} T^{2} \) |
| 19 | \( 1 + 214061380 T + p^{13} T^{2} \) |
| 23 | \( 1 + 830555544 T + p^{13} T^{2} \) |
| 29 | \( 1 - 1252400250 T + p^{13} T^{2} \) |
| 31 | \( 1 - 6159350552 T + p^{13} T^{2} \) |
| 37 | \( 1 + 5498191402 T + p^{13} T^{2} \) |
| 41 | \( 1 - 4678687878 T + p^{13} T^{2} \) |
| 43 | \( 1 - 7115013764 T + p^{13} T^{2} \) |
| 47 | \( 1 - 29528776992 T + p^{13} T^{2} \) |
| 53 | \( 1 - 204125042466 T + p^{13} T^{2} \) |
| 59 | \( 1 - 29909821020 T + p^{13} T^{2} \) |
| 61 | \( 1 + 134392006738 T + p^{13} T^{2} \) |
| 67 | \( 1 - 348518801948 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1314335409192 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1178875922326 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1072420659640 T + p^{13} T^{2} \) |
| 83 | \( 1 + 1124025139644 T + p^{13} T^{2} \) |
| 89 | \( 1 + 2235610909530 T + p^{13} T^{2} \) |
| 97 | \( 1 + 14215257165502 T + p^{13} T^{2} \) |
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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
−17.82361620189912140668522734249, −17.00302592986511598427319797782, −14.56381489529549369299404750502, −13.80819319585527486474627940091, −12.03921415167225390926982532099, −9.917920566413862741205682426969, −8.539747571839608944594943159390, −5.98566730249957353943830802997, −4.13943554132366905260912451026, −1.34421592758801776211863070896,
1.34421592758801776211863070896, 4.13943554132366905260912451026, 5.98566730249957353943830802997, 8.539747571839608944594943159390, 9.917920566413862741205682426969, 12.03921415167225390926982532099, 13.80819319585527486474627940091, 14.56381489529549369299404750502, 17.00302592986511598427319797782, 17.82361620189912140668522734249
