| L(s) = 1 | + 2-s + 3.08·3-s + 4-s + 0.522·5-s + 3.08·6-s + 0.722·7-s + 8-s + 6.54·9-s + 0.522·10-s − 1.07·11-s + 3.08·12-s + 3.41·13-s + 0.722·14-s + 1.61·15-s + 16-s + 2.46·17-s + 6.54·18-s + 6.72·19-s + 0.522·20-s + 2.23·21-s − 1.07·22-s − 9.00·23-s + 3.08·24-s − 4.72·25-s + 3.41·26-s + 10.9·27-s + 0.722·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·3-s + 0.5·4-s + 0.233·5-s + 1.26·6-s + 0.273·7-s + 0.353·8-s + 2.18·9-s + 0.165·10-s − 0.322·11-s + 0.891·12-s + 0.945·13-s + 0.193·14-s + 0.416·15-s + 0.250·16-s + 0.597·17-s + 1.54·18-s + 1.54·19-s + 0.116·20-s + 0.487·21-s − 0.228·22-s − 1.87·23-s + 0.630·24-s − 0.945·25-s + 0.668·26-s + 2.10·27-s + 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.688850115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.688850115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 - 0.522T + 5T^{2} \) |
| 7 | \( 1 - 0.722T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 + 9.00T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 - 0.278T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 5.78T + 53T^{2} \) |
| 59 | \( 1 - 4.73T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 9.26T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
| 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
−8.296027420041763056701698304361, −7.71730175824103975079246457306, −7.31727972137833695172281192817, −6.11034113855743398874603556576, −5.47795475082850390035914238362, −4.37416054145934987893409583460, −3.63095835447470232012197858669, −3.17343988685433672321242461048, −2.12059650107742108000408078488, −1.48326900373673475912690013753,
1.48326900373673475912690013753, 2.12059650107742108000408078488, 3.17343988685433672321242461048, 3.63095835447470232012197858669, 4.37416054145934987893409583460, 5.47795475082850390035914238362, 6.11034113855743398874603556576, 7.31727972137833695172281192817, 7.71730175824103975079246457306, 8.296027420041763056701698304361
