| L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s − 1.42·5-s + 0.470·6-s − 7-s − 1.77·8-s + 9-s − 0.672·10-s − 6.21·11-s − 1.77·12-s + 3.23·13-s − 0.470·14-s − 1.42·15-s + 2.71·16-s + 3.55·17-s + 0.470·18-s − 5.83·19-s + 2.53·20-s − 21-s − 2.92·22-s − 5.02·23-s − 1.77·24-s − 2.96·25-s + 1.52·26-s + 27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s − 0.638·5-s + 0.192·6-s − 0.377·7-s − 0.628·8-s + 0.333·9-s − 0.212·10-s − 1.87·11-s − 0.513·12-s + 0.898·13-s − 0.125·14-s − 0.368·15-s + 0.679·16-s + 0.862·17-s + 0.110·18-s − 1.33·19-s + 0.567·20-s − 0.218·21-s − 0.623·22-s − 1.04·23-s − 0.363·24-s − 0.592·25-s + 0.299·26-s + 0.192·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9250930048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9250930048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 11 | \( 1 + 6.21T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 + 4.64T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 + 9.55T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.98T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
| 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
−8.056752994525796348036574253564, −7.40300782325989970178593359068, −6.22430926043627971520669299564, −5.77684458004180621162637486399, −4.81922288708677669959175743014, −4.28874418490309272379407161918, −3.45936791834352636723877804252, −3.00734522770710446698947478393, −1.91114655779512634179836676047, −0.42326369135382850578600188778,
0.42326369135382850578600188778, 1.91114655779512634179836676047, 3.00734522770710446698947478393, 3.45936791834352636723877804252, 4.28874418490309272379407161918, 4.81922288708677669959175743014, 5.77684458004180621162637486399, 6.22430926043627971520669299564, 7.40300782325989970178593359068, 8.056752994525796348036574253564
