| L(s) = 1 | − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 4·13-s + 14-s + 16-s − 17-s + 7·19-s − 3·20-s + 4·25-s + 4·26-s − 28-s + 2·29-s + 7·31-s − 32-s + 34-s + 3·35-s + 5·37-s − 7·38-s + 3·40-s − 8·41-s + 43-s − 6·47-s − 6·49-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 − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s − 0.670·20-s + 4/5·25-s + 0.784·26-s − 0.188·28-s + 0.371·29-s + 1.25·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s + 0.821·37-s − 1.13·38-s + 0.474·40-s − 1.24·41-s + 0.152·43-s − 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.43713614427981, −14.63755343066100, −14.32023173742857, −13.54943554228684, −12.92335648942384, −12.28356680955403, −11.92005369181123, −11.46824210137184, −11.07626413964442, −10.20454091646022, −9.795767540310802, −9.408483047360131, −8.638138658769955, −7.977945326478958, −7.779964628719087, −7.151684544595135, −6.625535806020752, −6.015715878655276, −4.905311972138258, −4.792652327583246, −3.775617799249195, −3.151898453674887, −2.729243276916389, −1.660405302155060, −0.7402931338191690, 0,
0.7402931338191690, 1.660405302155060, 2.729243276916389, 3.151898453674887, 3.775617799249195, 4.792652327583246, 4.905311972138258, 6.015715878655276, 6.625535806020752, 7.151684544595135, 7.779964628719087, 7.977945326478958, 8.638138658769955, 9.408483047360131, 9.795767540310802, 10.20454091646022, 11.07626413964442, 11.46824210137184, 11.92005369181123, 12.28356680955403, 12.92335648942384, 13.54943554228684, 14.32023173742857, 14.63755343066100, 15.43713614427981
