| L(s) = 1 | − 1.30·2-s − 0.302·4-s − 7-s + 3·8-s + 3·11-s − 4.60·13-s + 1.30·14-s − 3.30·16-s − 2.60·17-s − 0.605·19-s − 3.90·22-s + 8.21·23-s + 6·26-s + 0.302·28-s + 0.394·29-s + 7.21·31-s − 1.69·32-s + 3.39·34-s − 10.2·37-s + 0.788·38-s − 2.39·43-s − 0.908·44-s − 10.6·46-s + 3.39·47-s + 49-s + 1.39·52-s − 11.2·53-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s − 0.377·7-s + 1.06·8-s + 0.904·11-s − 1.27·13-s + 0.348·14-s − 0.825·16-s − 0.631·17-s − 0.138·19-s − 0.833·22-s + 1.71·23-s + 1.17·26-s + 0.0572·28-s + 0.0732·29-s + 1.29·31-s − 0.300·32-s + 0.582·34-s − 1.67·37-s + 0.127·38-s − 0.365·43-s − 0.136·44-s − 1.57·46-s + 0.495·47-s + 0.142·49-s + 0.193·52-s − 1.53·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7809426925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7809426925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.277948927026265086072843465265, −8.877592668520726893469637519809, −7.977364365762281703477147644566, −7.04369091863314510300183569764, −6.58590204619360398955536186077, −5.12969305469015921393156494293, −4.52310423940609413112861753694, −3.34301704978670366630099398848, −2.05982445638797273093071187374, −0.71776872921996410606111884741,
0.71776872921996410606111884741, 2.05982445638797273093071187374, 3.34301704978670366630099398848, 4.52310423940609413112861753694, 5.12969305469015921393156494293, 6.58590204619360398955536186077, 7.04369091863314510300183569764, 7.977364365762281703477147644566, 8.877592668520726893469637519809, 9.277948927026265086072843465265
