| L(s) = 1 | + 5-s + 1.32·7-s − 5.32·11-s − 5.02·13-s + 6.34·17-s − 4.34·19-s + 1.70·23-s + 25-s + 29-s + 8.34·31-s + 1.32·35-s − 6.93·37-s + 1.02·41-s + 10.7·43-s + 0.679·47-s − 5.25·49-s − 2.38·53-s − 5.32·55-s − 10.4·59-s − 6.38·61-s − 5.02·65-s − 5.70·67-s + 3.61·71-s − 6.73·73-s − 7.02·77-s − 11.3·79-s + 3.96·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.499·7-s − 1.60·11-s − 1.39·13-s + 1.53·17-s − 0.997·19-s + 0.356·23-s + 0.200·25-s + 0.185·29-s + 1.49·31-s + 0.223·35-s − 1.14·37-s + 0.160·41-s + 1.63·43-s + 0.0990·47-s − 0.750·49-s − 0.327·53-s − 0.717·55-s − 1.35·59-s − 0.817·61-s − 0.623·65-s − 0.697·67-s + 0.428·71-s − 0.788·73-s − 0.800·77-s − 1.28·79-s + 0.434·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.679T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.79726546313300968099328794911, −7.35118778127331763399440308823, −6.34785680134332272186749732621, −5.51592119291581419662411843173, −5.01405023583652197521142122963, −4.34484038441904270340928245055, −2.95193618032618234049668460411, −2.54456662855585889951573625539, −1.42096228406856806185233043419, 0,
1.42096228406856806185233043419, 2.54456662855585889951573625539, 2.95193618032618234049668460411, 4.34484038441904270340928245055, 5.01405023583652197521142122963, 5.51592119291581419662411843173, 6.34785680134332272186749732621, 7.35118778127331763399440308823, 7.79726546313300968099328794911
