| L(s) = 1 | + 5-s − 3.60·7-s − 1.69·11-s − 13-s + 1.30·17-s + 5.90·19-s + 3·23-s + 25-s − 7.30·29-s + 2.39·31-s − 3.60·35-s + 37-s − 3·41-s − 0.0916·43-s − 0.394·47-s + 5.99·49-s − 1.69·55-s − 9.51·59-s + 4.21·61-s − 65-s + 14.5·67-s − 1.30·71-s − 7.51·73-s + 6.11·77-s + 2.90·79-s − 1.69·83-s + 1.30·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.36·7-s − 0.511·11-s − 0.277·13-s + 0.315·17-s + 1.35·19-s + 0.625·23-s + 0.200·25-s − 1.35·29-s + 0.430·31-s − 0.609·35-s + 0.164·37-s − 0.468·41-s − 0.0139·43-s − 0.0575·47-s + 0.857·49-s − 0.228·55-s − 1.23·59-s + 0.539·61-s − 0.124·65-s + 1.77·67-s − 0.154·71-s − 0.879·73-s + 0.697·77-s + 0.327·79-s − 0.186·83-s + 0.141·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 0.0916T + 43T^{2} \) |
| 47 | \( 1 + 0.394T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.51T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 1.30T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 - 2.90T + 79T^{2} \) |
| 83 | \( 1 + 1.69T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 1.60T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48712597815630369338010932878, −6.97715140747732091254888076982, −6.21259913193366900166703547021, −5.53276307788816685894213813648, −4.97177123663655742510742202726, −3.79736760262708240259401470599, −3.13577849875615004609689782273, −2.47948347419050031699091255548, −1.24176025117242012591182596277, 0,
1.24176025117242012591182596277, 2.47948347419050031699091255548, 3.13577849875615004609689782273, 3.79736760262708240259401470599, 4.97177123663655742510742202726, 5.53276307788816685894213813648, 6.21259913193366900166703547021, 6.97715140747732091254888076982, 7.48712597815630369338010932878
