| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·13-s − 14-s + 16-s − 8·17-s − 7·19-s − 7·23-s − 4·26-s − 28-s − 2·29-s − 2·31-s + 32-s − 8·34-s − 12·37-s − 7·38-s + 7·41-s − 4·43-s − 7·46-s + 2·47-s − 6·49-s − 4·52-s − 5·53-s − 56-s − 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s − 1.45·23-s − 0.784·26-s − 0.188·28-s − 0.371·29-s − 0.359·31-s + 0.176·32-s − 1.37·34-s − 1.97·37-s − 1.13·38-s + 1.09·41-s − 0.609·43-s − 1.03·46-s + 0.291·47-s − 6/7·49-s − 0.554·52-s − 0.686·53-s − 0.133·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.47573258140753, −14.09852015399591, −13.56058409667924, −12.88488610188543, −12.76957765987189, −12.19167406376312, −11.65737310164533, −11.05315402830027, −10.69068024532009, −10.09231193576852, −9.625434453983959, −8.910351668391808, −8.523150721984903, −7.872380753287046, −7.179914394999867, −6.785004243621004, −6.256652152618399, −5.825856883595197, −4.955584249139207, −4.610254619727646, −4.011306729612293, −3.522138118029200, −2.570856799310268, −2.184666358366814, −1.670345148872581, 0, 0,
1.670345148872581, 2.184666358366814, 2.570856799310268, 3.522138118029200, 4.011306729612293, 4.610254619727646, 4.955584249139207, 5.825856883595197, 6.256652152618399, 6.785004243621004, 7.179914394999867, 7.872380753287046, 8.523150721984903, 8.910351668391808, 9.625434453983959, 10.09231193576852, 10.69068024532009, 11.05315402830027, 11.65737310164533, 12.19167406376312, 12.76957765987189, 12.88488610188543, 13.56058409667924, 14.09852015399591, 14.47573258140753
