| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·13-s − 14-s + 16-s − 2·19-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 4·37-s + 2·38-s − 6·41-s + 2·43-s + 6·47-s + 49-s + 2·52-s − 6·53-s − 56-s + 6·58-s + 12·59-s − 8·61-s − 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.657·37-s + 0.324·38-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.277·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s − 1.02·61-s − 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.63319622044411, −11.99174484700406, −11.89168060216082, −11.18671985615824, −10.83617719885304, −10.57487897297173, −9.889955176658036, −9.556645994018411, −9.067822832847859, −8.585059899929971, −8.141687514979005, −7.844498783516714, −7.253164241608708, −6.784312555198120, −6.197533709175488, −5.982451936782519, −5.179072893544965, −4.864333763261160, −4.066912725854665, −3.755396392021307, −3.047441321655275, −2.437685692859473, −1.989746420988442, −1.295378893697587, −0.7984615264211220, 0,
0.7984615264211220, 1.295378893697587, 1.989746420988442, 2.437685692859473, 3.047441321655275, 3.755396392021307, 4.066912725854665, 4.864333763261160, 5.179072893544965, 5.982451936782519, 6.197533709175488, 6.784312555198120, 7.253164241608708, 7.844498783516714, 8.141687514979005, 8.585059899929971, 9.067822832847859, 9.556645994018411, 9.889955176658036, 10.57487897297173, 10.83617719885304, 11.18671985615824, 11.89168060216082, 11.99174484700406, 12.63319622044411
