| L(s) = 1 | + 2-s + 3-s + 4-s − 4.09·5-s + 6-s − 2.85·7-s + 8-s + 9-s − 4.09·10-s + 0.493·11-s + 12-s − 5.03·13-s − 2.85·14-s − 4.09·15-s + 16-s − 0.585·17-s + 18-s + 0.996·19-s − 4.09·20-s − 2.85·21-s + 0.493·22-s + 23-s + 24-s + 11.7·25-s − 5.03·26-s + 27-s − 2.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.82·5-s + 0.408·6-s − 1.07·7-s + 0.353·8-s + 0.333·9-s − 1.29·10-s + 0.148·11-s + 0.288·12-s − 1.39·13-s − 0.762·14-s − 1.05·15-s + 0.250·16-s − 0.141·17-s + 0.235·18-s + 0.228·19-s − 0.914·20-s − 0.622·21-s + 0.105·22-s + 0.208·23-s + 0.204·24-s + 2.34·25-s − 0.987·26-s + 0.192·27-s − 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.833068279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.833068279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 0.493T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 - 0.996T + 19T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 - 6.49T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 6.67T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 0.118T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
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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
−8.271007537483241312296107881368, −7.50850245948629598537655109300, −7.16469334894771380670589032414, −6.41376255720479517770213056365, −5.25855755585126495031644283423, −4.37663894612099892163713793325, −3.89938688455993292908169592663, −3.07649981055823145114303463744, −2.49966754385172427101149416578, −0.64937566553249103465873210922,
0.64937566553249103465873210922, 2.49966754385172427101149416578, 3.07649981055823145114303463744, 3.89938688455993292908169592663, 4.37663894612099892163713793325, 5.25855755585126495031644283423, 6.41376255720479517770213056365, 7.16469334894771380670589032414, 7.50850245948629598537655109300, 8.271007537483241312296107881368
