| L(s) = 1 | + 2-s + 1.73·3-s + 4-s − 5-s + 1.73·6-s − 1.89·7-s + 8-s + 0.00220·9-s − 10-s − 2.49·11-s + 1.73·12-s + 2.80·13-s − 1.89·14-s − 1.73·15-s + 16-s − 3.96·17-s + 0.00220·18-s − 6.34·19-s − 20-s − 3.28·21-s − 2.49·22-s + 5.04·23-s + 1.73·24-s + 25-s + 2.80·26-s − 5.19·27-s − 1.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.707·6-s − 0.716·7-s + 0.353·8-s + 0.000736·9-s − 0.316·10-s − 0.751·11-s + 0.500·12-s + 0.779·13-s − 0.506·14-s − 0.447·15-s + 0.250·16-s − 0.961·17-s + 0.000520·18-s − 1.45·19-s − 0.223·20-s − 0.716·21-s − 0.531·22-s + 1.05·23-s + 0.353·24-s + 0.200·25-s + 0.550·26-s − 0.999·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 + 7.16T + 31T^{2} \) |
| 37 | \( 1 - 5.77T + 37T^{2} \) |
| 41 | \( 1 + 8.93T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 0.375T + 59T^{2} \) |
| 61 | \( 1 + 8.97T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 + 6.37T + 97T^{2} \) |
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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
−8.076556952086785498945062337878, −7.37159673209929466892814961710, −6.56847758432266248670387098829, −5.92608621510275138210224878684, −4.92113200813425829800107979263, −4.07051322195073017103480455960, −3.38345113637988472347009439830, −2.73512803582452535536804666957, −1.86878461584591449843592520213, 0,
1.86878461584591449843592520213, 2.73512803582452535536804666957, 3.38345113637988472347009439830, 4.07051322195073017103480455960, 4.92113200813425829800107979263, 5.92608621510275138210224878684, 6.56847758432266248670387098829, 7.37159673209929466892814961710, 8.076556952086785498945062337878
