| L(s) = 1 | + (0.0691 − 1.31i)2-s + (1.21 − 0.985i)3-s + (0.253 + 0.0266i)4-s + (−0.928 + 2.03i)5-s + (−1.21 − 1.67i)6-s + (0.569 − 2.58i)7-s + (0.465 − 2.94i)8-s + (−0.113 + 0.536i)9-s + (2.61 + 1.36i)10-s + (0.468 − 0.0995i)11-s + (0.335 − 0.217i)12-s + (0.0993 − 0.195i)13-s + (−3.36 − 0.930i)14-s + (0.874 + 3.39i)15-s + (−3.34 − 0.712i)16-s + (−4.48 + 1.72i)17-s + ⋯ |
| L(s) = 1 | + (0.0488 − 0.932i)2-s + (0.702 − 0.568i)3-s + (0.126 + 0.0133i)4-s + (−0.415 + 0.909i)5-s + (−0.496 − 0.683i)6-s + (0.215 − 0.976i)7-s + (0.164 − 1.04i)8-s + (−0.0379 + 0.178i)9-s + (0.828 + 0.431i)10-s + (0.141 − 0.0300i)11-s + (0.0967 − 0.0628i)12-s + (0.0275 − 0.0540i)13-s + (−0.900 − 0.248i)14-s + (0.225 + 0.875i)15-s + (−0.837 − 0.178i)16-s + (−1.08 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0984 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0984 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11966 - 1.01439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11966 - 1.01439i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.928 - 2.03i)T \) |
| 7 | \( 1 + (-0.569 + 2.58i)T \) |
| good | 2 | \( 1 + (-0.0691 + 1.31i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.21 + 0.985i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-0.468 + 0.0995i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.0993 + 0.195i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (4.48 - 1.72i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.561 - 5.33i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.532 - 0.0278i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (2.22 - 3.06i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 3.05i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.15 - 6.39i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (2.67 - 0.870i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.03 + 6.03i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.68 - 6.98i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (3.04 + 3.76i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (3.58 - 3.97i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-6.61 + 5.95i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.65 + 12.1i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-3.80 - 2.76i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 7.14i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (3.76 + 8.46i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (9.38 + 1.48i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (3.91 + 4.35i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (9.51 - 1.50i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.47841457238126297234743660253, −11.27065423303392751673733034120, −10.78224748017914022928254267914, −9.821900921234564119923926582949, −8.193897279592724403797058353147, −7.37572557896693658407792498779, −6.49363847169130085422182418978, −4.14913559345846828433043188005, −3.09598681877467456717576306959, −1.79583302057052891125957213455,
2.54702590458905120205814054017, 4.37458025625311157148517628346, 5.40845577285403415592009071294, 6.68507343978791628535992436788, 7.980349346196537121763136748857, 8.855552035105303425174305913514, 9.335185230087903529924113755131, 11.16700694202696773859811726255, 11.89434643035056567933623955410, 13.10663367596706519060569665315
