| L(s) = 1 | − 2-s − 2.29·3-s + 4-s + 5-s + 2.29·6-s + 3.99·7-s − 8-s + 2.25·9-s − 10-s + 11-s − 2.29·12-s − 5.89·13-s − 3.99·14-s − 2.29·15-s + 16-s + 0.342·17-s − 2.25·18-s − 3.73·19-s + 20-s − 9.15·21-s − 22-s − 3.30·23-s + 2.29·24-s + 25-s + 5.89·26-s + 1.71·27-s + 3.99·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.32·3-s + 0.5·4-s + 0.447·5-s + 0.935·6-s + 1.51·7-s − 0.353·8-s + 0.750·9-s − 0.316·10-s + 0.301·11-s − 0.661·12-s − 1.63·13-s − 1.06·14-s − 0.591·15-s + 0.250·16-s + 0.0831·17-s − 0.530·18-s − 0.855·19-s + 0.223·20-s − 1.99·21-s − 0.213·22-s − 0.690·23-s + 0.467·24-s + 0.200·25-s + 1.15·26-s + 0.330·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 - 0.342T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 - 0.181T + 37T^{2} \) |
| 41 | \( 1 - 0.916T + 41T^{2} \) |
| 43 | \( 1 + 0.526T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−7.52746987359893637843820798813, −6.65693247770601290224511208959, −6.27882753116467080909465209885, −5.30558230289925881938514048312, −4.89579242501816557599503898619, −4.29156840626687348623767869788, −2.73695364698541065055155838674, −1.95159273437106382143558188586, −1.11615568650225888869257932308, 0,
1.11615568650225888869257932308, 1.95159273437106382143558188586, 2.73695364698541065055155838674, 4.29156840626687348623767869788, 4.89579242501816557599503898619, 5.30558230289925881938514048312, 6.27882753116467080909465209885, 6.65693247770601290224511208959, 7.52746987359893637843820798813
