| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 13-s − 14-s + 16-s + 17-s + 19-s − 20-s + 23-s + 25-s + 26-s + 28-s + 3·29-s − 10·31-s − 32-s − 34-s − 35-s + 10·37-s − 38-s + 40-s − 10·41-s − 6·43-s − 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 − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.229·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 0.557·29-s − 1.79·31-s − 0.176·32-s − 0.171·34-s − 0.169·35-s + 1.64·37-s − 0.162·38-s + 0.158·40-s − 1.56·41-s − 0.914·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037825276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037825276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.93541478838981, −12.38542699344589, −12.13191369131078, −11.43781812821244, −11.13045302763744, −10.83264194123055, −10.12831343476620, −9.619200056220610, −9.430289611782676, −8.655128876213587, −8.233018463281924, −7.974485623768773, −7.288302704867587, −6.972880528374889, −6.455237874056159, −5.765414597200116, −5.244127349056861, −4.807104389572643, −4.084707810825971, −3.560418277991330, −2.988410518478670, −2.364008877687704, −1.700998849482902, −1.133833210255559, −0.3447959591624547,
0.3447959591624547, 1.133833210255559, 1.700998849482902, 2.364008877687704, 2.988410518478670, 3.560418277991330, 4.084707810825971, 4.807104389572643, 5.244127349056861, 5.765414597200116, 6.455237874056159, 6.972880528374889, 7.288302704867587, 7.974485623768773, 8.233018463281924, 8.655128876213587, 9.430289611782676, 9.619200056220610, 10.12831343476620, 10.83264194123055, 11.13045302763744, 11.43781812821244, 12.13191369131078, 12.38542699344589, 12.93541478838981
