| L(s) = 1 | − 5-s + 4.46·7-s + 6.19·11-s − 6.73·13-s + 4.26·17-s − 2.73·19-s + 23-s + 25-s + 3.19·29-s − 31-s − 4.46·35-s + 9.39·37-s + 4.26·41-s − 3.46·43-s + 2.73·47-s + 12.9·49-s + 10.6·53-s − 6.19·55-s − 3.19·59-s − 7.26·61-s + 6.73·65-s + 5·67-s − 10.1·71-s − 14.7·73-s + 27.6·77-s + 4·79-s − 10.6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.68·7-s + 1.86·11-s − 1.86·13-s + 1.03·17-s − 0.626·19-s + 0.208·23-s + 0.200·25-s + 0.593·29-s − 0.179·31-s − 0.754·35-s + 1.54·37-s + 0.666·41-s − 0.528·43-s + 0.398·47-s + 1.84·49-s + 1.46·53-s − 0.835·55-s − 0.416·59-s − 0.930·61-s + 0.835·65-s + 0.610·67-s − 1.20·71-s − 1.72·73-s + 3.15·77-s + 0.450·79-s − 1.17·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.395439397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.395439397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.398394972117492602225408327999, −7.55491602449439102140022244576, −7.25851005391246496524177318065, −6.22161442695359963300102650094, −5.31749171100122395914654245978, −4.47178189430518470967501179811, −4.17806601028777823618010976529, −2.87801811757684074313834299640, −1.84670153597132589616586768874, −0.944740134060137521817513579406,
0.944740134060137521817513579406, 1.84670153597132589616586768874, 2.87801811757684074313834299640, 4.17806601028777823618010976529, 4.47178189430518470967501179811, 5.31749171100122395914654245978, 6.22161442695359963300102650094, 7.25851005391246496524177318065, 7.55491602449439102140022244576, 8.398394972117492602225408327999
