L(s) = 1 | − 1.23·2-s − 0.466·4-s + 1.86·5-s + 7-s + 3.05·8-s − 2.30·10-s + 0.846·11-s + 2.55·13-s − 1.23·14-s − 2.84·16-s + 7.07·17-s − 0.476·19-s − 0.869·20-s − 1.04·22-s + 23-s − 1.53·25-s − 3.15·26-s − 0.466·28-s − 8.63·29-s + 3.31·31-s − 2.58·32-s − 8.75·34-s + 1.86·35-s + 7.85·37-s + 0.590·38-s + 5.68·40-s − 2.82·41-s + ⋯ |
L(s) = 1 | − 0.875·2-s − 0.233·4-s + 0.832·5-s + 0.377·7-s + 1.07·8-s − 0.729·10-s + 0.255·11-s + 0.707·13-s − 0.330·14-s − 0.712·16-s + 1.71·17-s − 0.109·19-s − 0.194·20-s − 0.223·22-s + 0.208·23-s − 0.306·25-s − 0.619·26-s − 0.0881·28-s − 1.60·29-s + 0.594·31-s − 0.456·32-s − 1.50·34-s + 0.314·35-s + 1.29·37-s + 0.0957·38-s + 0.899·40-s − 0.441·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.278791265\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.278791265\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 11 | \( 1 - 0.846T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 0.476T + 19T^{2} \) |
| 29 | \( 1 + 8.63T + 29T^{2} \) |
| 31 | \( 1 - 3.31T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 0.274T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 + 1.66T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.31T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 - 8.76T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−9.422883358639061190128352310392, −8.943343270871149486677975678762, −7.908327513275514763081518216505, −7.50338857731832301090153144765, −6.16159102057667719226503694330, −5.53479416998887404198685294861, −4.49143852026301040558500622959, −3.43039159923026231303059761610, −1.92046405249632119270096136298, −1.01087951568853726922236975828,
1.01087951568853726922236975828, 1.92046405249632119270096136298, 3.43039159923026231303059761610, 4.49143852026301040558500622959, 5.53479416998887404198685294861, 6.16159102057667719226503694330, 7.50338857731832301090153144765, 7.908327513275514763081518216505, 8.943343270871149486677975678762, 9.422883358639061190128352310392