L(s) = 1 | − 2-s + 0.629·3-s + 4-s + 1.24·5-s − 0.629·6-s − 2.54·7-s − 8-s − 2.60·9-s − 1.24·10-s − 11-s + 0.629·12-s − 7.07·13-s + 2.54·14-s + 0.785·15-s + 16-s − 5.73·17-s + 2.60·18-s + 1.24·20-s − 1.60·21-s + 22-s + 1.76·23-s − 0.629·24-s − 3.44·25-s + 7.07·26-s − 3.52·27-s − 2.54·28-s − 5.47·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.363·3-s + 0.5·4-s + 0.557·5-s − 0.256·6-s − 0.961·7-s − 0.353·8-s − 0.867·9-s − 0.394·10-s − 0.301·11-s + 0.181·12-s − 1.96·13-s + 0.679·14-s + 0.202·15-s + 0.250·16-s − 1.39·17-s + 0.613·18-s + 0.278·20-s − 0.349·21-s + 0.213·22-s + 0.368·23-s − 0.128·24-s − 0.688·25-s + 1.38·26-s − 0.678·27-s − 0.480·28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5150526930\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5150526930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.629T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 + 5.47T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 0.519T + 37T^{2} \) |
| 41 | \( 1 - 0.911T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 - 4.00T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 + 17.0T + 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.71T + 89T^{2} \) |
| 97 | \( 1 - 5.06T + 97T^{2} \) |
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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
−7.76996807552450531784105758441, −7.31934632956243429542307610346, −6.50166745189790187144492956503, −5.95289995627360232927037042815, −5.16036906960103736370861254567, −4.28470584392315546702000376750, −3.12483971903438663272831999856, −2.53069452239049757476847540186, −2.03265134136195557396116568077, −0.35387914043484898965873833828,
0.35387914043484898965873833828, 2.03265134136195557396116568077, 2.53069452239049757476847540186, 3.12483971903438663272831999856, 4.28470584392315546702000376750, 5.16036906960103736370861254567, 5.95289995627360232927037042815, 6.50166745189790187144492956503, 7.31934632956243429542307610346, 7.76996807552450531784105758441