L(s) = 1 | + 2.61·2-s − 2·3-s + 4.85·4-s + 2.23·5-s − 5.23·6-s + 7-s + 7.47·8-s + 9-s + 5.85·10-s − 4.47·11-s − 9.70·12-s + 13-s + 2.61·14-s − 4.47·15-s + 9.85·16-s + 0.763·17-s + 2.61·18-s + 19-s + 10.8·20-s − 2·21-s − 11.7·22-s − 0.763·23-s − 14.9·24-s + 2.61·26-s + 4·27-s + 4.85·28-s + 5.23·29-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 1.15·3-s + 2.42·4-s + 0.999·5-s − 2.13·6-s + 0.377·7-s + 2.64·8-s + 0.333·9-s + 1.85·10-s − 1.34·11-s − 2.80·12-s + 0.277·13-s + 0.699·14-s − 1.15·15-s + 2.46·16-s + 0.185·17-s + 0.617·18-s + 0.229·19-s + 2.42·20-s − 0.436·21-s − 2.49·22-s − 0.159·23-s − 3.05·24-s + 0.513·26-s + 0.769·27-s + 0.917·28-s + 0.972·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.152635046\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.152635046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 0.763T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 7.47T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 5.23T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−11.54666126863449407821104118050, −10.67905984525184293819887547578, −10.09399422081914250266270420559, −8.181364651328660848323548833189, −6.83690835680249854199686308211, −6.11046736149924035799301969873, −5.20239239837318955223483704420, −4.98308498945101942555318352351, −3.27629903943302887299061169102, −1.96422743363950784130993968397,
1.96422743363950784130993968397, 3.27629903943302887299061169102, 4.98308498945101942555318352351, 5.20239239837318955223483704420, 6.11046736149924035799301969873, 6.83690835680249854199686308211, 8.181364651328660848323548833189, 10.09399422081914250266270420559, 10.67905984525184293819887547578, 11.54666126863449407821104118050