| L(s) = 1 | + 3-s + 1.73·7-s + 9-s − 1.73·11-s − 13-s − 4.46·17-s + 3.46·19-s + 1.73·21-s − 2·23-s + 27-s − 9.92·29-s + 1.19·31-s − 1.73·33-s + 7.46·37-s − 39-s − 9.46·41-s + 4.92·43-s − 10.6·47-s − 4·49-s − 4.46·51-s − 1.92·53-s + 3.46·57-s − 1.73·59-s − 5.53·61-s + 1.73·63-s − 7.73·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.654·7-s + 0.333·9-s − 0.522·11-s − 0.277·13-s − 1.08·17-s + 0.794·19-s + 0.377·21-s − 0.417·23-s + 0.192·27-s − 1.84·29-s + 0.214·31-s − 0.301·33-s + 1.22·37-s − 0.160·39-s − 1.47·41-s + 0.751·43-s − 1.55·47-s − 0.571·49-s − 0.625·51-s − 0.264·53-s + 0.458·57-s − 0.225·59-s − 0.708·61-s + 0.218·63-s − 0.944·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 9.92T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 1.92T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 + 5.53T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 - 6.39T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
| 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
−7.81931526795431775030148457717, −6.93076419460991232522066732014, −6.16917976653034522961243962019, −5.25818093304295091329302169154, −4.70976069931090867610643054847, −3.90137553403901883946351988471, −3.06265860163795337941528915294, −2.21420271529053899329789591262, −1.49714577594618275039195659693, 0,
1.49714577594618275039195659693, 2.21420271529053899329789591262, 3.06265860163795337941528915294, 3.90137553403901883946351988471, 4.70976069931090867610643054847, 5.25818093304295091329302169154, 6.16917976653034522961243962019, 6.93076419460991232522066732014, 7.81931526795431775030148457717
