| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 5·19-s − 20-s + 22-s − 6·23-s + 25-s + 26-s − 28-s − 6·29-s − 7·31-s + 32-s + 35-s + 2·37-s + 5·38-s − 40-s − 43-s + 44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 1.25·31-s + 0.176·32-s + 0.169·35-s + 0.328·37-s + 0.811·38-s − 0.158·40-s − 0.152·43-s + 0.150·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.38079999267964, −16.03345525310491, −15.38760112788563, −14.80018824753491, −14.36917494673821, −13.61209540247591, −13.31540999041309, −12.48756371501197, −12.15700967224522, −11.41527218929654, −11.11628244760406, −10.27544794768026, −9.641724706271393, −9.109154108772455, −8.242784455114606, −7.634936696537039, −7.104645197653749, −6.392140249805170, −5.705538162645430, −5.210963015623891, −4.247857085083773, −3.719939304750242, −3.174973933479356, −2.191882138181096, −1.308074811278908, 0,
1.308074811278908, 2.191882138181096, 3.174973933479356, 3.719939304750242, 4.247857085083773, 5.210963015623891, 5.705538162645430, 6.392140249805170, 7.104645197653749, 7.634936696537039, 8.242784455114606, 9.109154108772455, 9.641724706271393, 10.27544794768026, 11.11628244760406, 11.41527218929654, 12.15700967224522, 12.48756371501197, 13.31540999041309, 13.61209540247591, 14.36917494673821, 14.80018824753491, 15.38760112788563, 16.03345525310491, 16.38079999267964
