L(s) = 1 | − 2-s − 3-s + 4-s − 3.87·5-s + 6-s + 1.22·7-s − 8-s + 9-s + 3.87·10-s − 2.12·11-s − 12-s − 0.490·13-s − 1.22·14-s + 3.87·15-s + 16-s − 5.53·17-s − 18-s − 3.87·20-s − 1.22·21-s + 2.12·22-s − 8.94·23-s + 24-s + 10.0·25-s + 0.490·26-s − 27-s + 1.22·28-s + 8.47·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.73·5-s + 0.408·6-s + 0.463·7-s − 0.353·8-s + 0.333·9-s + 1.22·10-s − 0.639·11-s − 0.288·12-s − 0.135·13-s − 0.327·14-s + 1.00·15-s + 0.250·16-s − 1.34·17-s − 0.235·18-s − 0.867·20-s − 0.267·21-s + 0.452·22-s − 1.86·23-s + 0.204·24-s + 2.00·25-s + 0.0961·26-s − 0.192·27-s + 0.231·28-s + 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3347136152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3347136152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 0.490T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + 1.59T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 0.0418T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.768236281518707296709995962881, −8.253395467322348231704657581210, −7.66921702353050285878676493469, −6.94293886081660087225906776248, −6.14305302619220385256862163190, −4.87630038812921414582948533016, −4.32477722474393997363537319111, −3.27216467772833697452763983271, −1.98510688825886346194303371566, −0.41579777779247089120206764868,
0.41579777779247089120206764868, 1.98510688825886346194303371566, 3.27216467772833697452763983271, 4.32477722474393997363537319111, 4.87630038812921414582948533016, 6.14305302619220385256862163190, 6.94293886081660087225906776248, 7.66921702353050285878676493469, 8.253395467322348231704657581210, 8.768236281518707296709995962881