| L(s) = 1 | + 2·2-s + 2·4-s + 7-s + 3·13-s + 2·14-s − 4·16-s + 17-s − 7·19-s − 4·23-s + 6·26-s + 2·28-s − 8·29-s − 2·31-s − 8·32-s + 2·34-s + 11·37-s − 14·38-s − 9·41-s − 8·43-s − 8·46-s − 4·47-s + 49-s + 6·52-s − 5·53-s − 16·58-s − 4·59-s − 61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s + 0.832·13-s + 0.534·14-s − 16-s + 0.242·17-s − 1.60·19-s − 0.834·23-s + 1.17·26-s + 0.377·28-s − 1.48·29-s − 0.359·31-s − 1.41·32-s + 0.342·34-s + 1.80·37-s − 2.27·38-s − 1.40·41-s − 1.21·43-s − 1.17·46-s − 0.583·47-s + 1/7·49-s + 0.832·52-s − 0.686·53-s − 2.10·58-s − 0.520·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865833874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865833874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.14549443199688, −12.81968877021772, −12.31328310467466, −11.75672311787269, −11.28131730354804, −11.07823839833674, −10.45975572458082, −9.834300779688925, −9.354044711302508, −8.753995597666139, −8.212401065778086, −7.945264594367760, −7.103044407541174, −6.632513163667414, −6.129794569311715, −5.819567033424065, −5.206033258533576, −4.703722415251970, −4.175804863642651, −3.760531457582570, −3.312959989610624, −2.584125198597452, −1.935158041684369, −1.544306396237745, −0.3452718134885373,
0.3452718134885373, 1.544306396237745, 1.935158041684369, 2.584125198597452, 3.312959989610624, 3.760531457582570, 4.175804863642651, 4.703722415251970, 5.206033258533576, 5.819567033424065, 6.129794569311715, 6.632513163667414, 7.103044407541174, 7.945264594367760, 8.212401065778086, 8.753995597666139, 9.354044711302508, 9.834300779688925, 10.45975572458082, 11.07823839833674, 11.28131730354804, 11.75672311787269, 12.31328310467466, 12.81968877021772, 13.14549443199688
