| L(s) = 1 | + (0.980 + 0.198i)3-s + (0.969 + 0.246i)4-s + (−0.124 + 0.992i)7-s + (0.921 + 0.388i)9-s + (0.900 + 0.433i)12-s + (0.0270 + 0.0418i)13-s + (0.878 + 0.478i)16-s + (−0.733 − 0.680i)19-s + (−0.318 + 0.947i)21-s + (−0.921 − 0.388i)25-s + (0.826 + 0.563i)27-s + (−0.365 + 0.930i)28-s + (−0.583 − 1.01i)31-s + (0.797 + 0.603i)36-s + (−0.198 + 0.0148i)37-s + ⋯ |
| L(s) = 1 | + (0.980 + 0.198i)3-s + (0.969 + 0.246i)4-s + (−0.124 + 0.992i)7-s + (0.921 + 0.388i)9-s + (0.900 + 0.433i)12-s + (0.0270 + 0.0418i)13-s + (0.878 + 0.478i)16-s + (−0.733 − 0.680i)19-s + (−0.318 + 0.947i)21-s + (−0.921 − 0.388i)25-s + (0.826 + 0.563i)27-s + (−0.365 + 0.930i)28-s + (−0.583 − 1.01i)31-s + (0.797 + 0.603i)36-s + (−0.198 + 0.0148i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.136198020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.136198020\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.980 - 0.198i)T \) |
| 7 | \( 1 + (0.124 - 0.992i)T \) |
| 19 | \( 1 + (0.733 + 0.680i)T \) |
| good | 2 | \( 1 + (-0.969 - 0.246i)T^{2} \) |
| 5 | \( 1 + (0.921 + 0.388i)T^{2} \) |
| 11 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.0270 - 0.0418i)T + (-0.411 + 0.911i)T^{2} \) |
| 17 | \( 1 + (-0.853 - 0.521i)T^{2} \) |
| 23 | \( 1 + (0.0249 + 0.999i)T^{2} \) |
| 29 | \( 1 + (-0.661 + 0.749i)T^{2} \) |
| 31 | \( 1 + (0.583 + 1.01i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.198 - 0.0148i)T + (0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (0.797 - 0.603i)T^{2} \) |
| 43 | \( 1 + (0.952 + 0.518i)T + (0.542 + 0.840i)T^{2} \) |
| 47 | \( 1 + (0.995 - 0.0995i)T^{2} \) |
| 53 | \( 1 + (0.878 + 0.478i)T^{2} \) |
| 59 | \( 1 + (-0.542 - 0.840i)T^{2} \) |
| 61 | \( 1 + (-0.161 + 1.61i)T + (-0.980 - 0.198i)T^{2} \) |
| 67 | \( 1 + (-0.683 - 1.87i)T + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.318 + 0.947i)T^{2} \) |
| 73 | \( 1 + (1.62 - 0.831i)T + (0.583 - 0.811i)T^{2} \) |
| 79 | \( 1 + (-1.89 + 0.334i)T + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 89 | \( 1 + (-0.270 + 0.962i)T^{2} \) |
| 97 | \( 1 + (-1.46 + 1.22i)T + (0.173 - 0.984i)T^{2} \) |
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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
−8.945116770325284408150741606861, −8.344196115479831601393685084591, −7.68872346967118405512785456994, −6.87010364498684265746770859845, −6.16521814422477932240312307893, −5.24526441413688224950502446118, −4.12596266396609724233607593472, −3.28454874187309054005428458369, −2.40467007830147782828459560264, −1.89057465189217546754556394330,
1.35121036067675197241030400368, 2.13334786918665151624757216589, 3.27324632059321394074757259632, 3.82097613557405563968142039331, 4.92667685907402372843697986374, 6.13313061965022522637251821418, 6.72695355751455940733478145053, 7.49270199154642962549872523256, 7.916925399434914696134073967179, 8.830702761735940483395094339541
