| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 11-s + 12-s − 13-s − 2·14-s + 16-s − 17-s − 18-s + 2·21-s − 22-s + 4·23-s − 24-s + 26-s + 27-s + 2·28-s − 8·29-s + 8·31-s − 32-s + 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.48·29-s + 1.43·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.68109096222931, −12.37888306792051, −11.56940094917413, −11.33720343982427, −11.02958475360636, −10.35754395480639, −9.968081271665639, −9.541366362203288, −8.965669312242192, −8.701450020155068, −8.302983530495915, −7.623377763590155, −7.429753111066449, −6.923145150650363, −6.408714103104588, −5.731066945298915, −5.356248366425219, −4.590204842559987, −4.322370214326826, −3.557390321865278, −3.134456622636943, −2.280660321301229, −2.191139484337569, −1.302721297280108, −0.9288416130660150, 0,
0.9288416130660150, 1.302721297280108, 2.191139484337569, 2.280660321301229, 3.134456622636943, 3.557390321865278, 4.322370214326826, 4.590204842559987, 5.356248366425219, 5.731066945298915, 6.408714103104588, 6.923145150650363, 7.429753111066449, 7.623377763590155, 8.302983530495915, 8.701450020155068, 8.965669312242192, 9.541366362203288, 9.968081271665639, 10.35754395480639, 11.02958475360636, 11.33720343982427, 11.56940094917413, 12.37888306792051, 12.68109096222931
