| L(s) = 1 | + 2.66·3-s + 5-s + 1.57·7-s + 4.09·9-s + 0.654·11-s + 5.07·13-s + 2.66·15-s + 4.07·17-s − 0.594·19-s + 4.19·21-s + 0.919·23-s + 25-s + 2.92·27-s + 5.70·29-s + 3.25·31-s + 1.74·33-s + 1.57·35-s − 3.73·37-s + 13.5·39-s + 9.00·41-s − 10.7·43-s + 4.09·45-s − 13.2·47-s − 4.51·49-s + 10.8·51-s − 5.03·53-s + 0.654·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.447·5-s + 0.595·7-s + 1.36·9-s + 0.197·11-s + 1.40·13-s + 0.687·15-s + 0.988·17-s − 0.136·19-s + 0.916·21-s + 0.191·23-s + 0.200·25-s + 0.563·27-s + 1.05·29-s + 0.585·31-s + 0.303·33-s + 0.266·35-s − 0.613·37-s + 2.16·39-s + 1.40·41-s − 1.64·43-s + 0.611·45-s − 1.93·47-s − 0.645·49-s + 1.52·51-s − 0.691·53-s + 0.0882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.938027926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.938027926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 - 0.654T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 + 0.594T + 19T^{2} \) |
| 23 | \( 1 - 0.919T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 9.00T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 + 0.935T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 + 8.63T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 8.21T + 97T^{2} \) |
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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
−8.328653999013488593069385868726, −7.65501305942111684543018003304, −6.69878114009108552784934547877, −6.07412659665510952231824897505, −5.07719630243076870532874631282, −4.29525477227346079425209670144, −3.37269428761930493887275537127, −2.96131600341231616390584895158, −1.80616505460724956088905229242, −1.25728851882045693370630477095,
1.25728851882045693370630477095, 1.80616505460724956088905229242, 2.96131600341231616390584895158, 3.37269428761930493887275537127, 4.29525477227346079425209670144, 5.07719630243076870532874631282, 6.07412659665510952231824897505, 6.69878114009108552784934547877, 7.65501305942111684543018003304, 8.328653999013488593069385868726
