| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 2·11-s − 12-s − 4·13-s + 2·14-s + 16-s − 17-s − 18-s + 4·19-s + 2·21-s − 2·22-s − 2·23-s + 24-s + 4·26-s − 27-s − 2·28-s + 3·29-s − 31-s − 32-s − 2·33-s + 34-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.436·21-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−7.77723032247739794764171731597, −7.28525796307827527071911884686, −6.54792260211320558779566812777, −5.96726611979962733897055797920, −5.09441675150413277079140338661, −4.20627297604778543676272924260, −3.20629685249706834402851795617, −2.31240428645894660168264903155, −1.11744082962648240711342242022, 0,
1.11744082962648240711342242022, 2.31240428645894660168264903155, 3.20629685249706834402851795617, 4.20627297604778543676272924260, 5.09441675150413277079140338661, 5.96726611979962733897055797920, 6.54792260211320558779566812777, 7.28525796307827527071911884686, 7.77723032247739794764171731597
