| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 0.690·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 4·13-s − 0.690·14-s + 15-s + 16-s + 17-s − 18-s + 20-s + 0.690·21-s + 22-s − 8.90·23-s − 24-s + 25-s − 4·26-s + 27-s + 0.690·28-s − 0.690·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.260·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.184·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s + 0.150·21-s + 0.213·22-s − 1.85·23-s − 0.204·24-s + 0.200·25-s − 0.784·26-s + 0.192·27-s + 0.130·28-s − 0.128·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.111660588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.111660588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 0.690T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 + 0.690T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 9.59T + 37T^{2} \) |
| 41 | \( 1 + 7.59T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 7.59T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
| 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
−8.250142611572977791864408808721, −7.68910811180918737143910701642, −6.81477118317156525455305096045, −6.06099500308826895033894367984, −5.49075968901266940469697271108, −4.29492020614895836311207899761, −3.60049277446160549741587141088, −2.55457061955656138215221627372, −1.87769575839710081705941210228, −0.859596303813084407227525868747,
0.859596303813084407227525868747, 1.87769575839710081705941210228, 2.55457061955656138215221627372, 3.60049277446160549741587141088, 4.29492020614895836311207899761, 5.49075968901266940469697271108, 6.06099500308826895033894367984, 6.81477118317156525455305096045, 7.68910811180918737143910701642, 8.250142611572977791864408808721
