| L(s) = 1 | + 4·2-s + 16·4-s + 54·5-s + 104·7-s + 64·8-s + 216·10-s + 330·11-s − 46·13-s + 416·14-s + 256·16-s + 618·17-s + 361·19-s + 864·20-s + 1.32e3·22-s + 402·23-s − 209·25-s − 184·26-s + 1.66e3·28-s + 2.62e3·29-s − 2.36e3·31-s + 1.02e3·32-s + 2.47e3·34-s + 5.61e3·35-s − 1.21e4·37-s + 1.44e3·38-s + 3.45e3·40-s + 1.88e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.965·5-s + 0.802·7-s + 0.353·8-s + 0.683·10-s + 0.822·11-s − 0.0754·13-s + 0.567·14-s + 1/4·16-s + 0.518·17-s + 0.229·19-s + 0.482·20-s + 0.581·22-s + 0.158·23-s − 0.0668·25-s − 0.0533·26-s + 0.401·28-s + 0.580·29-s − 0.442·31-s + 0.176·32-s + 0.366·34-s + 0.774·35-s − 1.45·37-s + 0.162·38-s + 0.341·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.876807809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.876807809\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 104 T + p^{5} T^{2} \) |
| 11 | \( 1 - 30 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 618 T + p^{5} T^{2} \) |
| 23 | \( 1 - 402 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2628 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2368 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12130 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18864 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10408 T + p^{5} T^{2} \) |
| 47 | \( 1 - 4770 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19452 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30528 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11138 T + p^{5} T^{2} \) |
| 67 | \( 1 - 49508 T + p^{5} T^{2} \) |
| 71 | \( 1 + 7572 T + p^{5} T^{2} \) |
| 73 | \( 1 - 2342 T + p^{5} T^{2} \) |
| 79 | \( 1 - 22424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 46734 T + p^{5} T^{2} \) |
| 89 | \( 1 - 70104 T + p^{5} T^{2} \) |
| 97 | \( 1 - 105710 T + p^{5} T^{2} \) |
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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
−10.80314954226782226125677517767, −9.859706608272543043554854684048, −8.899525852228162904215314516226, −7.71924160064499133390242220788, −6.63319921564015299299007205400, −5.69068860135008260059566301013, −4.84359547032255771859246325878, −3.61970527924208798997624406449, −2.21313862164321987933051661464, −1.22804966424100054992964679446,
1.22804966424100054992964679446, 2.21313862164321987933051661464, 3.61970527924208798997624406449, 4.84359547032255771859246325878, 5.69068860135008260059566301013, 6.63319921564015299299007205400, 7.71924160064499133390242220788, 8.899525852228162904215314516226, 9.859706608272543043554854684048, 10.80314954226782226125677517767
