| L(s) = 1 | + 0.282·2-s − 0.0447·3-s − 1.92·4-s − 5-s − 0.0126·6-s + 0.565·7-s − 1.10·8-s − 2.99·9-s − 0.282·10-s + 3.86·11-s + 0.0858·12-s + 0.541·13-s + 0.159·14-s + 0.0447·15-s + 3.52·16-s + 3.38·17-s − 0.847·18-s − 1.46·19-s + 1.92·20-s − 0.0252·21-s + 1.09·22-s − 0.329·23-s + 0.0495·24-s + 25-s + 0.152·26-s + 0.268·27-s − 1.08·28-s + ⋯ |
| L(s) = 1 | + 0.199·2-s − 0.0258·3-s − 0.960·4-s − 0.447·5-s − 0.00516·6-s + 0.213·7-s − 0.391·8-s − 0.999·9-s − 0.0893·10-s + 1.16·11-s + 0.0247·12-s + 0.150·13-s + 0.0427·14-s + 0.0115·15-s + 0.881·16-s + 0.821·17-s − 0.199·18-s − 0.336·19-s + 0.429·20-s − 0.00551·21-s + 0.232·22-s − 0.0686·23-s + 0.0101·24-s + 0.200·25-s + 0.0300·26-s + 0.0516·27-s − 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.282T + 2T^{2} \) |
| 3 | \( 1 + 0.0447T + 3T^{2} \) |
| 7 | \( 1 - 0.565T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 0.541T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 0.329T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 9.28T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 - 5.10T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 9.55T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.38T + 97T^{2} \) |
| 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
−8.569508285877294866351302525601, −8.345621094034084130432094261298, −7.22985350853271475456341914393, −6.25025899182308371873474341990, −5.47366813593940667895071591291, −4.67756037507307452478220207225, −3.74990743046407002019556054085, −3.13901989128137591654787422856, −1.44995737430844028231390606681, 0,
1.44995737430844028231390606681, 3.13901989128137591654787422856, 3.74990743046407002019556054085, 4.67756037507307452478220207225, 5.47366813593940667895071591291, 6.25025899182308371873474341990, 7.22985350853271475456341914393, 8.345621094034084130432094261298, 8.569508285877294866351302525601
