L(s) = 1 | + 1.10·2-s + 3-s − 0.788·4-s + 2.79·5-s + 1.10·6-s − 1.33·7-s − 3.06·8-s + 9-s + 3.08·10-s − 0.596·11-s − 0.788·12-s + 1.71·13-s − 1.47·14-s + 2.79·15-s − 1.80·16-s − 17-s + 1.10·18-s − 2.45·19-s − 2.20·20-s − 1.33·21-s − 0.656·22-s + 0.281·23-s − 3.06·24-s + 2.83·25-s + 1.88·26-s + 27-s + 1.05·28-s + ⋯ |
L(s) = 1 | + 0.778·2-s + 0.577·3-s − 0.394·4-s + 1.25·5-s + 0.449·6-s − 0.505·7-s − 1.08·8-s + 0.333·9-s + 0.974·10-s − 0.179·11-s − 0.227·12-s + 0.474·13-s − 0.393·14-s + 0.722·15-s − 0.450·16-s − 0.242·17-s + 0.259·18-s − 0.562·19-s − 0.493·20-s − 0.292·21-s − 0.140·22-s + 0.0587·23-s − 0.626·24-s + 0.566·25-s + 0.369·26-s + 0.192·27-s + 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 0.596T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 + 2.90T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 - 1.53T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 + 0.215T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
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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
−7.35448845711518268117541712170, −6.54253424742924079130723521208, −6.04117924616884510379611799466, −5.33894902643317812656142773700, −4.76628170559568011393528818315, −3.71046939525949063231914543046, −3.33723129550006624481339655151, −2.32370409069405770534417730086, −1.61797003907202538300732764551, 0,
1.61797003907202538300732764551, 2.32370409069405770534417730086, 3.33723129550006624481339655151, 3.71046939525949063231914543046, 4.76628170559568011393528818315, 5.33894902643317812656142773700, 6.04117924616884510379611799466, 6.54253424742924079130723521208, 7.35448845711518268117541712170