| L(s) = 1 | + 16·4-s − 41·5-s + 49·7-s + 81·9-s + 169·13-s + 256·16-s − 97·19-s − 656·20-s + 967·23-s + 1.05e3·25-s + 784·28-s − 593·29-s + 1.10e3·31-s − 2.00e3·35-s + 1.29e3·36-s − 2.46e3·41-s − 3.67e3·43-s − 3.32e3·45-s + 2.14e3·47-s + 2.40e3·49-s + 2.70e3·52-s − 5.39e3·53-s + 1.13e3·59-s + 3.96e3·63-s + 4.09e3·64-s − 6.92e3·65-s − 9.81e3·73-s + ⋯ |
| L(s) = 1 | + 4-s − 1.63·5-s + 7-s + 9-s + 13-s + 16-s − 0.268·19-s − 1.63·20-s + 1.82·23-s + 1.68·25-s + 28-s − 0.705·29-s + 1.14·31-s − 1.63·35-s + 36-s − 1.46·41-s − 1.98·43-s − 1.63·45-s + 0.970·47-s + 49-s + 52-s − 1.91·53-s + 0.326·59-s + 63-s + 64-s − 1.63·65-s − 1.84·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.981973556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981973556\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( 1 + 41 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 97 T + p^{4} T^{2} \) |
| 23 | \( 1 - 967 T + p^{4} T^{2} \) |
| 29 | \( 1 + 593 T + p^{4} T^{2} \) |
| 31 | \( 1 - 1103 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 + 2462 T + p^{4} T^{2} \) |
| 43 | \( 1 + 3673 T + p^{4} T^{2} \) |
| 47 | \( 1 - 2143 T + p^{4} T^{2} \) |
| 53 | \( 1 + 5393 T + p^{4} T^{2} \) |
| 59 | \( 1 - 1138 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 9817 T + p^{4} T^{2} \) |
| 79 | \( 1 + 7993 T + p^{4} T^{2} \) |
| 83 | \( 1 - 11503 T + p^{4} T^{2} \) |
| 89 | \( 1 - 11383 T + p^{4} T^{2} \) |
| 97 | \( 1 + 1657 T + p^{4} T^{2} \) |
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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
−13.16840616219324410329285153402, −11.97413198773251448819492189267, −11.31440826791628746603171969843, −10.53773123289976608087096860117, −8.537818357251341560942923479708, −7.63692052627089201536134423019, −6.75063452462684851341265838120, −4.75939178165769056180722698816, −3.44096392440973820212473172703, −1.32005647531832515120755152816,
1.32005647531832515120755152816, 3.44096392440973820212473172703, 4.75939178165769056180722698816, 6.75063452462684851341265838120, 7.63692052627089201536134423019, 8.537818357251341560942923479708, 10.53773123289976608087096860117, 11.31440826791628746603171969843, 11.97413198773251448819492189267, 13.16840616219324410329285153402
