L(s) = 1 | + (0.832 + 1.44i)2-s + (0.579 + 1.00i)3-s + (−0.385 + 0.667i)4-s + (−0.5 − 0.866i)5-s + (−0.965 + 1.67i)6-s − 2.43·7-s + 2.04·8-s + (0.827 − 1.43i)9-s + (0.832 − 1.44i)10-s − 5.75·11-s − 0.893·12-s + (0.797 − 1.38i)13-s + (−2.02 − 3.51i)14-s + (0.579 − 1.00i)15-s + (2.47 + 4.28i)16-s + (2.99 + 5.18i)17-s + ⋯ |
L(s) = 1 | + (0.588 + 1.01i)2-s + (0.334 + 0.579i)3-s + (−0.192 + 0.333i)4-s + (−0.223 − 0.387i)5-s + (−0.394 + 0.682i)6-s − 0.920·7-s + 0.723·8-s + (0.275 − 0.477i)9-s + (0.263 − 0.455i)10-s − 1.73·11-s − 0.258·12-s + (0.221 − 0.383i)13-s + (−0.541 − 0.938i)14-s + (0.149 − 0.259i)15-s + (0.618 + 1.07i)16-s + (0.725 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05943 + 0.825342i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05943 + 0.825342i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.149 + 4.35i)T \) |
good | 2 | \( 1 + (-0.832 - 1.44i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.579 - 1.00i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + (-0.797 + 1.38i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.470 + 0.814i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + (-3.15 - 5.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.26 + 3.93i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.47 - 7.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.09 + 1.90i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.39 - 9.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 + 9.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.504 - 0.874i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.41 + 7.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.12 - 8.87i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.80 + 6.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.11T + 83T^{2} \) |
| 89 | \( 1 + (-5.55 + 9.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.02 + 3.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.54874534009600698051124125936, −13.05637004340446750175659203340, −12.82337044684863547482026097445, −10.74531488682028142385503912449, −9.896683940117095071802750474267, −8.485288031526507491745496813211, −7.33746146797625876524125585321, −6.01627479357606074512520245157, −4.88222841823786374027609550257, −3.43145695284455644466045054413,
2.34737899186682024585389260596, 3.46472210352320493045315886827, 5.24124426899623972660220834703, 7.15562095679816148750554807057, 7.942396577047318941995325308265, 9.879168573822033094588934840321, 10.66642836378644564513874016978, 11.85561052462833769707490500790, 12.86768598469082587225251727402, 13.37415616752082970205865156648