| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s − 2·10-s + 11-s + 2·12-s − 4·13-s + 2·14-s − 15-s − 4·16-s + 6·17-s + 2·18-s − 2·20-s + 21-s + 2·22-s + 4·23-s − 4·25-s − 8·26-s + 27-s + 2·28-s + 6·29-s − 2·30-s − 31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s − 16-s + 1.45·17-s + 0.471·18-s − 0.447·20-s + 0.218·21-s + 0.426·22-s + 0.834·23-s − 4/5·25-s − 1.56·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.365·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.291465330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.291465330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.86912949168740, −13.67763587096339, −12.99282750173080, −12.37640860183431, −12.18263556297328, −11.78708315018382, −11.15765244892032, −10.61477681748572, −9.755842098856777, −9.667895372895115, −8.830680867509656, −8.299685773448251, −7.738662294185707, −7.288237014330826, −6.734460576775812, −6.136576470705674, −5.434454423046023, −4.949707574136988, −4.629214314369737, −3.783619827857404, −3.544799895316230, −2.837637603302289, −2.351135292346627, −1.531937206981055, −0.6048062931342043,
0.6048062931342043, 1.531937206981055, 2.351135292346627, 2.837637603302289, 3.544799895316230, 3.783619827857404, 4.629214314369737, 4.949707574136988, 5.434454423046023, 6.136576470705674, 6.734460576775812, 7.288237014330826, 7.738662294185707, 8.299685773448251, 8.830680867509656, 9.667895372895115, 9.755842098856777, 10.61477681748572, 11.15765244892032, 11.78708315018382, 12.18263556297328, 12.37640860183431, 12.99282750173080, 13.67763587096339, 13.86912949168740
