| L(s) = 1 | − 3-s − 7-s + 9-s + 5.41·11-s − 4.34·13-s − 1.07·17-s + 4.34·19-s + 21-s − 6.34·23-s − 27-s − 8.83·29-s − 4.34·31-s − 5.41·33-s + 8.68·37-s + 4.34·39-s + 8.34·41-s − 6.15·43-s + 6.83·47-s + 49-s + 1.07·51-s − 6.18·53-s − 4.34·57-s − 6.83·59-s − 4.52·61-s − 63-s + 6.34·69-s − 14.0·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 0.333·9-s + 1.63·11-s − 1.20·13-s − 0.261·17-s + 0.995·19-s + 0.218·21-s − 1.32·23-s − 0.192·27-s − 1.64·29-s − 0.779·31-s − 0.943·33-s + 1.42·37-s + 0.694·39-s + 1.30·41-s − 0.938·43-s + 0.997·47-s + 0.142·49-s + 0.151·51-s − 0.849·53-s − 0.574·57-s − 0.890·59-s − 0.579·61-s − 0.125·63-s + 0.763·69-s − 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 0.680T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
| 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
−7.69293959077972303293268548239, −7.42291578674046365785240599840, −6.41821501354579640339154606050, −5.97838545750475262426520244071, −5.08595788298688738686373603200, −4.19665787990758347454261913576, −3.59216739070791161437655696357, −2.36015385608100650414984528349, −1.33179347039825393918165419925, 0,
1.33179347039825393918165419925, 2.36015385608100650414984528349, 3.59216739070791161437655696357, 4.19665787990758347454261913576, 5.08595788298688738686373603200, 5.97838545750475262426520244071, 6.41821501354579640339154606050, 7.42291578674046365785240599840, 7.69293959077972303293268548239
