| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 2·11-s − 12-s − 5·13-s + 16-s − 17-s − 2·18-s − 19-s − 2·22-s − 24-s − 5·26-s + 5·27-s + 29-s − 7·31-s + 32-s + 2·33-s − 34-s − 2·36-s − 8·37-s − 38-s + 5·39-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.603·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.229·19-s − 0.426·22-s − 0.204·24-s − 0.980·26-s + 0.962·27-s + 0.185·29-s − 1.25·31-s + 0.176·32-s + 0.348·33-s − 0.171·34-s − 1/3·36-s − 1.31·37-s − 0.162·38-s + 0.800·39-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5976298896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5976298896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.82416093574976, −14.17695467661211, −13.73072623033210, −13.18844096828078, −12.51058127759703, −12.20045055937904, −11.77067174706528, −11.14911727777323, −10.58897414086061, −10.29432258870145, −9.533980822787431, −8.876814128695662, −8.325650505515897, −7.579159846494043, −7.139542800815496, −6.562946858959124, −5.906005506710388, −5.319626741500475, −4.993404561286017, −4.414902851115677, −3.513105610819508, −2.944969030411972, −2.290404003463824, −1.588254408065096, −0.2406390115399461,
0.2406390115399461, 1.588254408065096, 2.290404003463824, 2.944969030411972, 3.513105610819508, 4.414902851115677, 4.993404561286017, 5.319626741500475, 5.906005506710388, 6.562946858959124, 7.139542800815496, 7.579159846494043, 8.325650505515897, 8.876814128695662, 9.533980822787431, 10.29432258870145, 10.58897414086061, 11.14911727777323, 11.77067174706528, 12.20045055937904, 12.51058127759703, 13.18844096828078, 13.73072623033210, 14.17695467661211, 14.82416093574976
