| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 14-s + 15-s − 16-s + 17-s − 18-s − 4·19-s + 20-s − 21-s − 4·22-s − 3·24-s + 25-s − 27-s − 28-s − 6·29-s − 30-s − 5·32-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 + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.78778584408156, −12.35277306053217, −11.97118574422363, −11.42315735037551, −11.11561399515353, −10.52133026669833, −10.25025270740830, −9.652336075358165, −9.171359536765995, −8.837507559593371, −8.250202831741577, −8.034608218589339, −7.284458559610051, −6.898299055224978, −6.551321876548401, −5.792694220196197, −5.257322102818102, −4.896451809990824, −4.231677917282429, −3.811599365276798, −3.539339076821344, −2.461731588715832, −1.682607818683707, −1.412884456788701, −0.6074734876820264, 0,
0.6074734876820264, 1.412884456788701, 1.682607818683707, 2.461731588715832, 3.539339076821344, 3.811599365276798, 4.231677917282429, 4.896451809990824, 5.257322102818102, 5.792694220196197, 6.551321876548401, 6.898299055224978, 7.284458559610051, 8.034608218589339, 8.250202831741577, 8.837507559593371, 9.171359536765995, 9.652336075358165, 10.25025270740830, 10.52133026669833, 11.11561399515353, 11.42315735037551, 11.97118574422363, 12.35277306053217, 12.78778584408156
