L(s) = 1 | − 0.628·2-s + 3-s − 1.60·4-s − 0.0316·5-s − 0.628·6-s − 0.526·7-s + 2.26·8-s + 9-s + 0.0198·10-s − 2.67·11-s − 1.60·12-s + 13-s + 0.330·14-s − 0.0316·15-s + 1.78·16-s + 2.49·17-s − 0.628·18-s + 5.54·19-s + 0.0508·20-s − 0.526·21-s + 1.67·22-s − 3.20·23-s + 2.26·24-s − 4.99·25-s − 0.628·26-s + 27-s + 0.845·28-s + ⋯ |
L(s) = 1 | − 0.444·2-s + 0.577·3-s − 0.802·4-s − 0.0141·5-s − 0.256·6-s − 0.199·7-s + 0.800·8-s + 0.333·9-s + 0.00629·10-s − 0.806·11-s − 0.463·12-s + 0.277·13-s + 0.0883·14-s − 0.00817·15-s + 0.447·16-s + 0.605·17-s − 0.148·18-s + 1.27·19-s + 0.0113·20-s − 0.114·21-s + 0.358·22-s − 0.669·23-s + 0.462·24-s − 0.999·25-s − 0.123·26-s + 0.192·27-s + 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.349084914\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.349084914\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 0.628T + 2T^{2} \) |
| 5 | \( 1 + 0.0316T + 5T^{2} \) |
| 7 | \( 1 + 0.526T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 1.08T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 9.34T + 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
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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
−8.303194805494122201150360987621, −7.899360736486284931118469077574, −7.38538626363942367410795770693, −6.20342473931129543095753340566, −5.39547094017278511368567478063, −4.65905662924231996257554760022, −3.73428397297336136365520325601, −3.05808715576880931796639780914, −1.86519964455614887181383202717, −0.70679882112069791193999060271,
0.70679882112069791193999060271, 1.86519964455614887181383202717, 3.05808715576880931796639780914, 3.73428397297336136365520325601, 4.65905662924231996257554760022, 5.39547094017278511368567478063, 6.20342473931129543095753340566, 7.38538626363942367410795770693, 7.899360736486284931118469077574, 8.303194805494122201150360987621