| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3.35·7-s + 8-s + 9-s − 0.962·11-s − 12-s + 1.61·13-s + 3.35·14-s + 16-s + 0.387·17-s + 18-s + 19-s − 3.35·21-s − 0.962·22-s + 0.962·23-s − 24-s + 1.61·26-s − 27-s + 3.35·28-s + 6.96·29-s + 3.35·31-s + 32-s + 0.962·33-s + 0.387·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.26·7-s + 0.353·8-s + 0.333·9-s − 0.290·11-s − 0.288·12-s + 0.447·13-s + 0.895·14-s + 0.250·16-s + 0.0940·17-s + 0.235·18-s + 0.229·19-s − 0.731·21-s − 0.205·22-s + 0.200·23-s − 0.204·24-s + 0.316·26-s − 0.192·27-s + 0.633·28-s + 1.29·29-s + 0.601·31-s + 0.176·32-s + 0.167·33-s + 0.0665·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.948551418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.948551418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 0.962T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 0.387T + 17T^{2} \) |
| 23 | \( 1 - 0.962T + 23T^{2} \) |
| 29 | \( 1 - 6.96T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 0.962T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.22T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
| 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
−8.388894630390848685379926718386, −8.157547731331686439011018941587, −7.04385114077478658360195358489, −6.47082808670588982150472077549, −5.44789255013430167252404435544, −4.99160121048984436468009542127, −4.26807159652247261969913956316, −3.23510892995052702924737950843, −2.07330895612808480526761291781, −1.05493677737871007217103639281,
1.05493677737871007217103639281, 2.07330895612808480526761291781, 3.23510892995052702924737950843, 4.26807159652247261969913956316, 4.99160121048984436468009542127, 5.44789255013430167252404435544, 6.47082808670588982150472077549, 7.04385114077478658360195358489, 8.157547731331686439011018941587, 8.388894630390848685379926718386
