| L(s) = 1 | − 1.30·2-s − 0.302·4-s + 3.60·5-s + 7-s + 3·8-s − 4.69·10-s + 0.605·13-s − 1.30·14-s − 3.30·16-s + 6.30·17-s + 3·19-s − 1.09·20-s + 6.30·23-s + 7.99·25-s − 0.788·26-s − 0.302·28-s + 8.30·29-s + 31-s − 1.69·32-s − 8.21·34-s + 3.60·35-s + 9.21·37-s − 3.90·38-s + 10.8·40-s + 7·41-s − 5.30·43-s − 8.21·46-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s + 1.61·5-s + 0.377·7-s + 1.06·8-s − 1.48·10-s + 0.167·13-s − 0.348·14-s − 0.825·16-s + 1.52·17-s + 0.688·19-s − 0.244·20-s + 1.31·23-s + 1.59·25-s − 0.154·26-s − 0.0572·28-s + 1.54·29-s + 0.179·31-s − 0.300·32-s − 1.40·34-s + 0.609·35-s + 1.51·37-s − 0.634·38-s + 1.71·40-s + 1.09·41-s − 0.808·43-s − 1.21·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.128641541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.128641541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.697T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 - 0.697T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−7.961083481976143769329879098664, −7.39038235903413019365567364335, −6.48919547613271431608966067873, −5.83262485204259101669740118669, −5.07452459652330767978626851481, −4.61983898418802152104477359507, −3.28368868629168618940077015733, −2.50116160886032842779820104546, −1.34774191974826322371713560314, −1.03729060068105822185530304431,
1.03729060068105822185530304431, 1.34774191974826322371713560314, 2.50116160886032842779820104546, 3.28368868629168618940077015733, 4.61983898418802152104477359507, 5.07452459652330767978626851481, 5.83262485204259101669740118669, 6.48919547613271431608966067873, 7.39038235903413019365567364335, 7.961083481976143769329879098664
