| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 3·11-s + 13-s − 14-s + 16-s + 5·17-s − 19-s + 20-s + 3·22-s − 23-s + 25-s + 26-s − 28-s + 29-s − 4·31-s + 32-s + 5·34-s − 35-s + 4·37-s − 38-s + 40-s + 5·41-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 + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 0.208·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s + 0.857·34-s − 0.169·35-s + 0.657·37-s − 0.162·38-s + 0.158·40-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.291737203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.291737203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.40722564438054, −15.81907123172982, −15.04277398217933, −14.65885749220164, −14.05864693228342, −13.64273637122932, −12.96645887672072, −12.40580308127025, −11.97287364729478, −11.30058992500315, −10.64110333895749, −10.05191240271175, −9.428230848373879, −8.875365114830575, −8.041431548943177, −7.341433715397539, −6.725693202249663, −5.983030645875989, −5.698196227469469, −4.790527678680367, −4.013777060104623, −3.454698099698731, −2.662484432414028, −1.751828865684346, −0.8847411991995838,
0.8847411991995838, 1.751828865684346, 2.662484432414028, 3.454698099698731, 4.013777060104623, 4.790527678680367, 5.698196227469469, 5.983030645875989, 6.725693202249663, 7.341433715397539, 8.041431548943177, 8.875365114830575, 9.428230848373879, 10.05191240271175, 10.64110333895749, 11.30058992500315, 11.97287364729478, 12.40580308127025, 12.96645887672072, 13.64273637122932, 14.05864693228342, 14.65885749220164, 15.04277398217933, 15.81907123172982, 16.40722564438054
