| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s − 13-s + 4·14-s + 16-s − 17-s + 18-s + 2·19-s − 4·21-s + 22-s − 24-s − 26-s − 27-s + 4·28-s − 6·29-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 + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.213·22-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-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 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.55089181335320, −12.27488182533760, −11.78106796597090, −11.45335977101676, −11.12948083220973, −10.60914182635491, −10.26646280579916, −9.522519870450280, −9.257337354421746, −8.442531550460307, −8.039942189700306, −7.752252405186998, −7.090076911456127, −6.645227773488032, −6.284079256783595, −5.501078921719667, −5.170049773825811, −4.929088192009698, −4.325639208963494, −3.811898630151182, −3.377853988237144, −2.332642564049252, −2.231855622646196, −1.331422544974703, −1.030881580578906, 0,
1.030881580578906, 1.331422544974703, 2.231855622646196, 2.332642564049252, 3.377853988237144, 3.811898630151182, 4.325639208963494, 4.929088192009698, 5.170049773825811, 5.501078921719667, 6.284079256783595, 6.645227773488032, 7.090076911456127, 7.752252405186998, 8.039942189700306, 8.442531550460307, 9.257337354421746, 9.522519870450280, 10.26646280579916, 10.60914182635491, 11.12948083220973, 11.45335977101676, 11.78106796597090, 12.27488182533760, 12.55089181335320
