| L(s) = 1 | − 2.56·3-s + 0.728·5-s + 3.51·7-s + 3.58·9-s + 2.48·11-s − 13-s − 1.86·15-s − 2.68·17-s + 2.82·19-s − 9.00·21-s + 1.17·23-s − 4.46·25-s − 1.50·27-s + 29-s + 7.55·31-s − 6.37·33-s + 2.55·35-s − 1.38·37-s + 2.56·39-s + 1.18·41-s + 7.86·43-s + 2.61·45-s − 4.29·47-s + 5.32·49-s + 6.88·51-s + 8.82·53-s + 1.80·55-s + ⋯ |
| L(s) = 1 | − 1.48·3-s + 0.325·5-s + 1.32·7-s + 1.19·9-s + 0.748·11-s − 0.277·13-s − 0.482·15-s − 0.650·17-s + 0.648·19-s − 1.96·21-s + 0.244·23-s − 0.893·25-s − 0.289·27-s + 0.185·29-s + 1.35·31-s − 1.10·33-s + 0.432·35-s − 0.228·37-s + 0.410·39-s + 0.184·41-s + 1.19·43-s + 0.389·45-s − 0.626·47-s + 0.760·49-s + 0.964·51-s + 1.21·53-s + 0.243·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552487860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552487860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 0.728T + 5T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 1.88T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.094223655289658817126971086977, −7.07664300611723831807840146647, −6.69358107833476254762601021555, −5.70547266017249263314282013864, −5.39900672088867960868404229212, −4.54537951956190350407996915937, −4.06874935189632407862044230340, −2.58049789896029677737233585562, −1.56653560292715220160810202950, −0.76322462555446435012707954690,
0.76322462555446435012707954690, 1.56653560292715220160810202950, 2.58049789896029677737233585562, 4.06874935189632407862044230340, 4.54537951956190350407996915937, 5.39900672088867960868404229212, 5.70547266017249263314282013864, 6.69358107833476254762601021555, 7.07664300611723831807840146647, 8.094223655289658817126971086977
