L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 4·13-s − 2·14-s + 16-s + 2·17-s + 8·19-s − 20-s + 8·23-s + 25-s − 4·26-s + 2·28-s + 4·29-s − 31-s − 32-s − 2·34-s − 2·35-s − 12·37-s − 8·38-s + 40-s + 10·41-s − 8·43-s − 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s + 0.742·29-s − 0.179·31-s − 0.176·32-s − 0.342·34-s − 0.338·35-s − 1.97·37-s − 1.29·38-s + 0.158·40-s + 1.56·41-s − 1.21·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 337590 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337590 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.54759058037384, −12.27147620432722, −11.84399424442650, −11.22763058415779, −11.09264494811849, −10.59548836387505, −10.17121289345266, −9.389156898179592, −9.215052345034999, −8.710274904146741, −8.089854364670824, −7.941911932911028, −7.298138392941224, −6.967363585654655, −6.427482581718833, −5.781222173116623, −5.198183656443115, −4.992950523122834, −4.216147967853053, −3.588230706627729, −3.069206905639366, −2.802431741233571, −1.700877021884514, −1.290685862484188, −0.9552041914066435, 0,
0.9552041914066435, 1.290685862484188, 1.700877021884514, 2.802431741233571, 3.069206905639366, 3.588230706627729, 4.216147967853053, 4.992950523122834, 5.198183656443115, 5.781222173116623, 6.427482581718833, 6.967363585654655, 7.298138392941224, 7.941911932911028, 8.089854364670824, 8.710274904146741, 9.215052345034999, 9.389156898179592, 10.17121289345266, 10.59548836387505, 11.09264494811849, 11.22763058415779, 11.84399424442650, 12.27147620432722, 12.54759058037384