| L(s) = 1 | − 1.73·3-s + 2.27·11-s + 6.09·13-s − 4.77·17-s − 4.27·19-s − 0.894·23-s + 5.19·27-s − 3.27·29-s − 4.27·31-s − 3.94·33-s + 5.61·37-s − 10.5·39-s − 11.2·41-s + 6.50·43-s − 2.15·47-s + 8.27·51-s − 7.40·53-s + 7.40·57-s + 4.27·59-s + 1.54·61-s + 13.9·67-s + 1.54·69-s + 10.5·71-s − 2.15·73-s − 0.274·79-s − 9·81-s − 5.67·83-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.685·11-s + 1.68·13-s − 1.15·17-s − 0.980·19-s − 0.186·23-s + 1.00·27-s − 0.608·29-s − 0.767·31-s − 0.685·33-s + 0.923·37-s − 1.68·39-s − 1.76·41-s + 0.992·43-s − 0.313·47-s + 1.15·51-s − 1.01·53-s + 0.980·57-s + 0.556·59-s + 0.198·61-s + 1.69·67-s + 0.186·69-s + 1.25·71-s − 0.251·73-s − 0.0309·79-s − 81-s − 0.622·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 + 0.894T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 6.50T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 + 7.40T + 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 - 1.54T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 + 0.274T + 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−8.005280960767961524229883662537, −6.79469678303608685571050275035, −6.44011141723905205047133193548, −5.86170155937473186147973948889, −5.05210379333635105301566586495, −4.15171451778313039751111306220, −3.55993157071463785888079402183, −2.24380582052460664552484634844, −1.21905613991825137378597035857, 0,
1.21905613991825137378597035857, 2.24380582052460664552484634844, 3.55993157071463785888079402183, 4.15171451778313039751111306220, 5.05210379333635105301566586495, 5.86170155937473186147973948889, 6.44011141723905205047133193548, 6.79469678303608685571050275035, 8.005280960767961524229883662537
