| L(s) = 1 | − 12·3-s + 54·5-s − 88·7-s − 99·9-s + 540·11-s − 418·13-s − 648·15-s + 594·17-s + 836·19-s + 1.05e3·21-s − 4.10e3·23-s − 209·25-s + 4.10e3·27-s − 594·29-s + 4.25e3·31-s − 6.48e3·33-s − 4.75e3·35-s − 298·37-s + 5.01e3·39-s + 1.72e4·41-s − 1.21e4·43-s − 5.34e3·45-s − 1.29e3·47-s − 9.06e3·49-s − 7.12e3·51-s + 1.94e4·53-s + 2.91e4·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.965·5-s − 0.678·7-s − 0.407·9-s + 1.34·11-s − 0.685·13-s − 0.743·15-s + 0.498·17-s + 0.531·19-s + 0.522·21-s − 1.61·23-s − 0.0668·25-s + 1.08·27-s − 0.131·29-s + 0.795·31-s − 1.03·33-s − 0.655·35-s − 0.0357·37-s + 0.528·39-s + 1.60·41-s − 0.997·43-s − 0.393·45-s − 0.0855·47-s − 0.539·49-s − 0.383·51-s + 0.953·53-s + 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7805376825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7805376825\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 + 88 T + p^{5} T^{2} \) |
| 11 | \( 1 - 540 T + p^{5} T^{2} \) |
| 13 | \( 1 + 418 T + p^{5} T^{2} \) |
| 17 | \( 1 - 594 T + p^{5} T^{2} \) |
| 19 | \( 1 - 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 594 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 + 298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 + 122398 T + p^{5} T^{2} \) |
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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
−24.66474956351914942067996001362, −22.76638104045062086910631624859, −21.82648947337359762085007791626, −19.75976565483922320580773419775, −17.71694741483574255598828058356, −16.59946943253973234810046665847, −14.11508352828821016305212270051, −11.99964135123060776621941064917, −9.747199643115818989680697413553, −6.10006048288389002479771045385,
6.10006048288389002479771045385, 9.747199643115818989680697413553, 11.99964135123060776621941064917, 14.11508352828821016305212270051, 16.59946943253973234810046665847, 17.71694741483574255598828058356, 19.75976565483922320580773419775, 21.82648947337359762085007791626, 22.76638104045062086910631624859, 24.66474956351914942067996001362
