| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s + 11-s + 2·14-s − 4·16-s + 6·17-s + 7·19-s − 2·20-s − 2·22-s + 7·23-s − 4·25-s − 2·28-s + 7·29-s − 31-s + 8·32-s − 12·34-s + 35-s − 8·37-s − 14·38-s − 2·41-s − 5·43-s + 2·44-s − 14·46-s + 5·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s + 0.301·11-s + 0.534·14-s − 16-s + 1.45·17-s + 1.60·19-s − 0.447·20-s − 0.426·22-s + 1.45·23-s − 4/5·25-s − 0.377·28-s + 1.29·29-s − 0.179·31-s + 1.41·32-s − 2.05·34-s + 0.169·35-s − 1.31·37-s − 2.27·38-s − 0.312·41-s − 0.762·43-s + 0.301·44-s − 2.06·46-s + 0.729·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184438451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184438451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.63143571004397, −13.15433447647864, −12.32484910279735, −12.06243571655873, −11.53702086270991, −11.16011614542554, −10.40008168407489, −10.06638004379260, −9.754917825949290, −9.146846315849759, −8.705494998457495, −8.273583010225129, −7.615937735807907, −7.334435519569020, −6.888795865718554, −6.278143832469856, −5.444673126044293, −5.114610712624530, −4.374651326431469, −3.493286532346481, −3.265080932013045, −2.488699085264539, −1.567236228270874, −1.049043049730235, −0.5197845112627575,
0.5197845112627575, 1.049043049730235, 1.567236228270874, 2.488699085264539, 3.265080932013045, 3.493286532346481, 4.374651326431469, 5.114610712624530, 5.444673126044293, 6.278143832469856, 6.888795865718554, 7.334435519569020, 7.615937735807907, 8.273583010225129, 8.705494998457495, 9.146846315849759, 9.754917825949290, 10.06638004379260, 10.40008168407489, 11.16011614542554, 11.53702086270991, 12.06243571655873, 12.32484910279735, 13.15433447647864, 13.63143571004397
