| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (0.959 + 0.281i)6-s + (0.208 − 0.133i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (0.108 − 0.751i)11-s + (−0.142 + 0.989i)12-s + (−2.45 − 1.57i)13-s + (0.162 + 0.187i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (6.73 + 1.97i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (0.391 + 0.115i)6-s + (0.0787 − 0.0505i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.266 + 0.170i)10-s + (0.0325 − 0.226i)11-s + (−0.0410 + 0.285i)12-s + (−0.679 − 0.436i)13-s + (0.0433 + 0.0499i)14-s + (−0.107 − 0.234i)15-s + (0.210 − 0.135i)16-s + (1.63 + 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68508 - 0.279937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68508 - 0.279937i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-0.715 + 4.74i)T \) |
| good | 7 | \( 1 + (-0.208 + 0.133i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.108 + 0.751i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.45 + 1.57i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.73 - 1.97i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.70 + 2.26i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (7.35 + 2.16i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.32 + 2.89i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.244 + 0.282i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-8.38 + 9.67i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.50 - 3.30i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + (-8.29 + 5.32i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-11.5 - 7.42i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.29 - 2.82i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.45 - 10.0i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.519 - 3.61i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (14.9 - 4.39i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (7.14 + 4.59i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.22 - 8.34i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.09 - 2.40i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (7.63 - 8.81i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.12191031104702209924410788587, −9.469369360485857229957530581121, −8.524563226795090172998458390351, −7.63572164112312723495381622336, −7.15100923573424366043364987284, −5.75053484567468201993955752521, −5.38148891505155648487192399680, −3.93691097332328091763712399044, −2.70745767650587013185429217445, −0.968215996376873470784493456519,
1.55107223424734163427681813199, 2.97460753781719545396765662129, 3.68436137455617720191866086541, 5.08273404448781091529894056899, 5.61502054119617894917770619239, 7.21705100242746854091875479381, 7.890694394468702170018032195978, 9.334758879952425010133399127489, 9.637865587318878096311755030738, 10.34742250206344677716507051506
