| L(s) = 1 | + (0.0270 − 0.516i)2-s + (−1.27 + 1.03i)3-s + (1.72 + 0.181i)4-s + (−0.509 + 2.17i)5-s + (0.499 + 0.687i)6-s + (−2.33 + 1.24i)7-s + (0.302 − 1.90i)8-s + (−0.0630 + 0.296i)9-s + (1.11 + 0.322i)10-s + (1.94 − 0.412i)11-s + (−2.38 + 1.54i)12-s + (−2.78 + 5.46i)13-s + (0.578 + 1.24i)14-s + (−1.60 − 3.30i)15-s + (2.41 + 0.512i)16-s + (6.82 − 2.62i)17-s + ⋯ |
| L(s) = 1 | + (0.0191 − 0.365i)2-s + (−0.736 + 0.596i)3-s + (0.861 + 0.0905i)4-s + (−0.227 + 0.973i)5-s + (0.203 + 0.280i)6-s + (−0.882 + 0.469i)7-s + (0.106 − 0.674i)8-s + (−0.0210 + 0.0988i)9-s + (0.351 + 0.101i)10-s + (0.585 − 0.124i)11-s + (−0.688 + 0.447i)12-s + (−0.772 + 1.51i)13-s + (0.154 + 0.331i)14-s + (−0.413 − 0.853i)15-s + (0.602 + 0.128i)16-s + (1.65 − 0.635i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.818669 + 0.560816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.818669 + 0.560816i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.509 - 2.17i)T \) |
| 7 | \( 1 + (2.33 - 1.24i)T \) |
| good | 2 | \( 1 + (-0.0270 + 0.516i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (1.27 - 1.03i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.94 + 0.412i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.78 - 5.46i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-6.82 + 2.62i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.302 + 2.87i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-4.32 - 0.226i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (2.61 - 3.60i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.199 + 0.447i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (3.06 + 4.71i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-1.18 + 0.384i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.03 + 2.03i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.09 + 2.84i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-4.38 - 5.40i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-9.92 + 11.0i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (8.11 - 7.30i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.00505 - 0.0131i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (0.558 + 0.405i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 7.28i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 9.40i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.44 + 1.02i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (1.52 + 1.69i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-4.31 + 0.683i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.40083275555979077017773987775, −11.71639259632231815237510024106, −11.08632606664375526075392119691, −10.11447873636936709548816104396, −9.326873835250235135723832265024, −7.30943766830375946966929651601, −6.68144714918192248297784202646, −5.48048506318083715507438527773, −3.76080201057333088255866410274, −2.53541390637865761505146931784,
1.05581242342103216654867651731, 3.40791739487613286868124083293, 5.39086041924798530414422158201, 6.10462232423042253355161296790, 7.25850863104145649767538722157, 8.037936778757126141299533787142, 9.645330218710896828966330361684, 10.58280639504854506924022294831, 11.94183757438961990513490250540, 12.36416862652112242513314393933
