| L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 11-s − 2·12-s − 15-s + 4·16-s − 17-s + 5·19-s + 2·20-s + 21-s − 4·23-s + 25-s + 27-s − 2·28-s − 9·29-s − 33-s − 35-s − 2·36-s − 4·37-s + 5·41-s + 43-s + 2·44-s − 45-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.242·17-s + 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 1.67·29-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 0.657·37-s + 0.780·41-s + 0.152·43-s + 0.301·44-s − 0.149·45-s − 1.75·47-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.90081861527836, −12.59324381648123, −11.99331523508558, −11.65448322012802, −11.02844030687174, −10.59726186115897, −10.08584113499891, −9.514041654939382, −9.265313515088763, −8.824632714688217, −8.205315318517375, −7.878347964539633, −7.517071474587783, −7.097146904425361, −6.198229925506512, −5.838172519205895, −5.072230514930659, −4.911907211841976, −4.233577064507074, −3.731809344576758, −3.373076824210828, −2.791947432725921, −1.949803215906675, −1.508436547429354, −0.6767254631548338, 0,
0.6767254631548338, 1.508436547429354, 1.949803215906675, 2.791947432725921, 3.373076824210828, 3.731809344576758, 4.233577064507074, 4.911907211841976, 5.072230514930659, 5.838172519205895, 6.198229925506512, 7.097146904425361, 7.517071474587783, 7.878347964539633, 8.205315318517375, 8.824632714688217, 9.265313515088763, 9.514041654939382, 10.08584113499891, 10.59726186115897, 11.02844030687174, 11.65448322012802, 11.99331523508558, 12.59324381648123, 12.90081861527836
