L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 13-s + 14-s + 16-s − 17-s + 7·19-s + 20-s − 3·23-s + 25-s + 26-s − 28-s + 8·29-s − 5·31-s − 32-s + 34-s − 35-s + 5·37-s − 7·38-s − 40-s + 10·41-s − 2·43-s + 3·46-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 + 1.60·19-s + 0.223·20-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.821·37-s − 1.13·38-s − 0.158·40-s + 1.56·41-s − 0.304·43-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.29218076294302, −12.93863088998048, −12.26041941963679, −12.01563784910667, −11.39926320530833, −10.99288973504852, −10.42260548533930, −9.956503012525374, −9.495076662843770, −9.363712042458931, −8.569769422755587, −8.220802052507465, −7.538309919052986, −7.250446523709562, −6.637802007942487, −6.150162816333632, −5.661543292353209, −5.148257901919941, −4.519730376376319, −3.828919557491445, −3.230955811460451, −2.601247396609877, −2.242690150179831, −1.311638607372081, −0.8912923620656970, 0,
0.8912923620656970, 1.311638607372081, 2.242690150179831, 2.601247396609877, 3.230955811460451, 3.828919557491445, 4.519730376376319, 5.148257901919941, 5.661543292353209, 6.150162816333632, 6.637802007942487, 7.250446523709562, 7.538309919052986, 8.220802052507465, 8.569769422755587, 9.363712042458931, 9.495076662843770, 9.956503012525374, 10.42260548533930, 10.99288973504852, 11.39926320530833, 12.01563784910667, 12.26041941963679, 12.93863088998048, 13.29218076294302