L(s) = 1 | + 1.58·2-s + 0.522·4-s + 3.92·5-s − 0.0296·7-s − 2.34·8-s + 6.24·10-s + 5.94·11-s − 4.56·13-s − 0.0470·14-s − 4.77·16-s + 3.85·17-s + 5.00·19-s + 2.05·20-s + 9.43·22-s − 7.53·23-s + 10.4·25-s − 7.25·26-s − 0.0154·28-s + 1.68·29-s + 1.82·31-s − 2.88·32-s + 6.11·34-s − 0.116·35-s + 11.2·37-s + 7.94·38-s − 9.22·40-s − 4.38·41-s + ⋯ |
L(s) = 1 | + 1.12·2-s + 0.261·4-s + 1.75·5-s − 0.0111·7-s − 0.829·8-s + 1.97·10-s + 1.79·11-s − 1.26·13-s − 0.0125·14-s − 1.19·16-s + 0.934·17-s + 1.14·19-s + 0.459·20-s + 2.01·22-s − 1.57·23-s + 2.08·25-s − 1.42·26-s − 0.00292·28-s + 0.313·29-s + 0.327·31-s − 0.510·32-s + 1.04·34-s − 0.0196·35-s + 1.85·37-s + 1.28·38-s − 1.45·40-s − 0.685·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.798090238\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.798090238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 5 | \( 1 - 3.92T + 5T^{2} \) |
| 7 | \( 1 + 0.0296T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 - 0.889T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 0.811T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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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
−9.641360397597583779044011445026, −9.217143464845159652379191013614, −7.922534476920766696738273411821, −6.60264116905674815750694800858, −6.21306916890091527086739512741, −5.37494348748016907383447674312, −4.68769431000180427237404269561, −3.57709195956647098867717854370, −2.57443631297472011743549759926, −1.43046217139591197499567287492,
1.43046217139591197499567287492, 2.57443631297472011743549759926, 3.57709195956647098867717854370, 4.68769431000180427237404269561, 5.37494348748016907383447674312, 6.21306916890091527086739512741, 6.60264116905674815750694800858, 7.922534476920766696738273411821, 9.217143464845159652379191013614, 9.641360397597583779044011445026