| L(s) = 1 | + 2·2-s − 8.39·3-s + 4·4-s + 1.25·5-s − 16.7·6-s + 8·8-s + 43.4·9-s + 2.50·10-s + 34.5·11-s − 33.5·12-s + 19.9·13-s − 10.4·15-s + 16·16-s − 12.2·17-s + 86.8·18-s − 149.·19-s + 5.00·20-s + 69.0·22-s + 23·23-s − 67.1·24-s − 123.·25-s + 39.9·26-s − 138.·27-s + 38.5·29-s − 20.9·30-s − 71.0·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.111·5-s − 1.14·6-s + 0.353·8-s + 1.60·9-s + 0.0790·10-s + 0.946·11-s − 0.807·12-s + 0.425·13-s − 0.180·15-s + 0.250·16-s − 0.174·17-s + 1.13·18-s − 1.80·19-s + 0.0559·20-s + 0.669·22-s + 0.208·23-s − 0.571·24-s − 0.987·25-s + 0.300·26-s − 0.983·27-s + 0.246·29-s − 0.127·30-s − 0.411·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 8.39T + 27T^{2} \) |
| 5 | \( 1 - 1.25T + 125T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 149.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 38.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 217.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 671.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 16.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 93.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 15.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 760.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 852.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 131.T + 9.12e5T^{2} \) |
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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
−8.172181889247873671319201453514, −7.06767970565633565576157327903, −6.36604724965318712135729564390, −6.06555523318789143338555797205, −5.15859196756317326825155013453, −4.35330781321526751776575008847, −3.74977584355336041936692457368, −2.19876995775920328806401422627, −1.18104278346053763862240017287, 0,
1.18104278346053763862240017287, 2.19876995775920328806401422627, 3.74977584355336041936692457368, 4.35330781321526751776575008847, 5.15859196756317326825155013453, 6.06555523318789143338555797205, 6.36604724965318712135729564390, 7.06767970565633565576157327903, 8.172181889247873671319201453514
