| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 2·13-s + 14-s + 15-s + 16-s + 7·17-s − 18-s + 2·19-s − 20-s + 21-s − 5·22-s + 7·23-s + 24-s + 25-s + 2·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s − 1.06·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.66161741131835, −12.28668613835123, −12.00994759776961, −11.65979893112371, −11.06995141738792, −10.72043472930065, −10.09018860525757, −9.746251695945651, −9.354246670036675, −8.825301870206093, −8.445569667312107, −7.604223760529559, −7.462233177186135, −6.926408005657252, −6.432591488271199, −6.032348252302329, −5.429124944961386, −4.812163983506659, −4.457957829164799, −3.517917426254737, −3.317081740246215, −2.777475666917185, −1.750419354421157, −1.225464743770817, −0.8370722807723463, 0,
0.8370722807723463, 1.225464743770817, 1.750419354421157, 2.777475666917185, 3.317081740246215, 3.517917426254737, 4.457957829164799, 4.812163983506659, 5.429124944961386, 6.032348252302329, 6.432591488271199, 6.926408005657252, 7.462233177186135, 7.604223760529559, 8.445569667312107, 8.825301870206093, 9.354246670036675, 9.746251695945651, 10.09018860525757, 10.72043472930065, 11.06995141738792, 11.65979893112371, 12.00994759776961, 12.28668613835123, 12.66161741131835
