| L(s) = 1 | + 2-s + 4-s + 0.622·5-s − 7-s + 8-s + 0.622·10-s − 2.42·11-s + 2.62·13-s − 14-s + 16-s + 4.42·17-s − 19-s + 0.622·20-s − 2.42·22-s + 1.37·23-s − 4.61·25-s + 2.62·26-s − 28-s + 9.61·29-s + 3.80·31-s + 32-s + 4.42·34-s − 0.622·35-s + 5.80·37-s − 38-s + 0.622·40-s + 0.755·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.278·5-s − 0.377·7-s + 0.353·8-s + 0.196·10-s − 0.732·11-s + 0.727·13-s − 0.267·14-s + 0.250·16-s + 1.07·17-s − 0.229·19-s + 0.139·20-s − 0.517·22-s + 0.287·23-s − 0.922·25-s + 0.514·26-s − 0.188·28-s + 1.78·29-s + 0.683·31-s + 0.176·32-s + 0.759·34-s − 0.105·35-s + 0.954·37-s − 0.162·38-s + 0.0983·40-s + 0.118·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997438767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997438767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.622T + 5T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 - 0.755T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 - 0.949T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 + 4.42T + 67T^{2} \) |
| 71 | \( 1 + 0.857T + 71T^{2} \) |
| 73 | \( 1 - 6.85T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
| 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
−8.932208571769634668987973674859, −8.049589851917529919871114972992, −7.42951301828567622090344125061, −6.32200208763900613786059847005, −5.93617529873925963653732477395, −5.01999485051464308101566694265, −4.15625128870022072196939321939, −3.18622998278885204491576428880, −2.43703269449703157678076425390, −1.04598084507985906819486555199,
1.04598084507985906819486555199, 2.43703269449703157678076425390, 3.18622998278885204491576428880, 4.15625128870022072196939321939, 5.01999485051464308101566694265, 5.93617529873925963653732477395, 6.32200208763900613786059847005, 7.42951301828567622090344125061, 8.049589851917529919871114972992, 8.932208571769634668987973674859
