| L(s) = 1 | − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 3·11-s + 5·13-s + 14-s + 16-s + 2·19-s − 3·20-s + 3·22-s + 4·25-s − 5·26-s − 28-s − 2·31-s − 32-s + 3·35-s + 37-s − 2·38-s + 3·40-s + 5·43-s − 3·44-s − 6·47-s + 49-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.904·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.670·20-s + 0.639·22-s + 4/5·25-s − 0.980·26-s − 0.188·28-s − 0.359·31-s − 0.176·32-s + 0.507·35-s + 0.164·37-s − 0.324·38-s + 0.474·40-s + 0.762·43-s − 0.452·44-s − 0.875·47-s + 1/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.41962157320885, −14.91207276981289, −14.06406084176228, −13.61400780138979, −12.94101748089378, −12.42225686149266, −12.01382903257318, −11.20877016410542, −10.97345236909093, −10.63044458350093, −9.656516163966809, −9.394845877747765, −8.555865130577941, −8.094430063790571, −7.847967299781328, −7.135860187110628, −6.597362126054434, −5.925629145078652, −5.288309470594533, −4.476288286881544, −3.770453706001840, −3.301396400175027, −2.661187637067047, −1.639973778132689, −0.7789336074930774, 0,
0.7789336074930774, 1.639973778132689, 2.661187637067047, 3.301396400175027, 3.770453706001840, 4.476288286881544, 5.288309470594533, 5.925629145078652, 6.597362126054434, 7.135860187110628, 7.847967299781328, 8.094430063790571, 8.555865130577941, 9.394845877747765, 9.656516163966809, 10.63044458350093, 10.97345236909093, 11.20877016410542, 12.01382903257318, 12.42225686149266, 12.94101748089378, 13.61400780138979, 14.06406084176228, 14.91207276981289, 15.41962157320885
