L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 3·13-s + 14-s − 15-s + 16-s − 8·17-s − 18-s + 20-s + 21-s + 22-s + 24-s + 25-s + 3·26-s − 27-s − 28-s − 6·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.27462191887328, −12.96582050811520, −12.48074833341874, −11.89368385109446, −11.46604055323060, −10.87231853329931, −10.76115061408718, −9.946458299319876, −9.686041269210627, −9.302776615659789, −8.665062309614916, −8.257848145729030, −7.561755742767492, −7.105654096177703, −6.657248278655553, −6.241329840284212, −5.689015544115453, −5.139778216522308, −4.516788684741007, −4.118644754856024, −3.195663567696570, −2.575566828381041, −2.137802229565546, −1.513306240527450, −0.5851465156272514, 0,
0.5851465156272514, 1.513306240527450, 2.137802229565546, 2.575566828381041, 3.195663567696570, 4.118644754856024, 4.516788684741007, 5.139778216522308, 5.689015544115453, 6.241329840284212, 6.657248278655553, 7.105654096177703, 7.561755742767492, 8.257848145729030, 8.665062309614916, 9.302776615659789, 9.686041269210627, 9.946458299319876, 10.76115061408718, 10.87231853329931, 11.46604055323060, 11.89368385109446, 12.48074833341874, 12.96582050811520, 13.27462191887328