| L(s) = 1 | − 1.56·3-s − 1.56·5-s + 0.438·7-s − 0.561·9-s + 5.12·11-s + 2.43·15-s + 3.56·17-s − 2·19-s − 0.684·21-s − 3.12·23-s − 2.56·25-s + 5.56·27-s − 5.12·29-s + 5.12·31-s − 8·33-s − 0.684·35-s − 9.56·37-s − 8·41-s + 9.56·43-s + 0.876·45-s − 7.56·47-s − 6.80·49-s − 5.56·51-s + 12.2·53-s − 8·55-s + 3.12·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.901·3-s − 0.698·5-s + 0.165·7-s − 0.187·9-s + 1.54·11-s + 0.629·15-s + 0.863·17-s − 0.458·19-s − 0.149·21-s − 0.651·23-s − 0.512·25-s + 1.07·27-s − 0.951·29-s + 0.920·31-s − 1.39·33-s − 0.115·35-s − 1.57·37-s − 1.24·41-s + 1.45·43-s + 0.130·45-s − 1.10·47-s − 0.972·49-s − 0.778·51-s + 1.68·53-s − 1.07·55-s + 0.413·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014376903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014376903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 9.56T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 + 6.68T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 8.24T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−8.696771018842542207263351075491, −8.160674307231356457011931974376, −7.17257939692592664729694541977, −6.50716171125282464990685070156, −5.78926185900397933972520926536, −5.00465887704030808227982046069, −4.01441604485184811788092879351, −3.44526309235531808256193394657, −1.90127515572272056260445980288, −0.66267037626513989326510550280,
0.66267037626513989326510550280, 1.90127515572272056260445980288, 3.44526309235531808256193394657, 4.01441604485184811788092879351, 5.00465887704030808227982046069, 5.78926185900397933972520926536, 6.50716171125282464990685070156, 7.17257939692592664729694541977, 8.160674307231356457011931974376, 8.696771018842542207263351075491
