| L(s) = 1 | + 2-s + 4-s + 1.84·5-s + 2.73·7-s + 8-s + 1.84·10-s − 3.84·11-s − 1.70·13-s + 2.73·14-s + 16-s + 1.33·17-s + 4.04·19-s + 1.84·20-s − 3.84·22-s − 3.24·23-s − 1.60·25-s − 1.70·26-s + 2.73·28-s + 1.77·29-s − 3.60·31-s + 32-s + 1.33·34-s + 5.02·35-s − 4.85·37-s + 4.04·38-s + 1.84·40-s + 11.5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.823·5-s + 1.03·7-s + 0.353·8-s + 0.582·10-s − 1.16·11-s − 0.472·13-s + 0.729·14-s + 0.250·16-s + 0.324·17-s + 0.928·19-s + 0.411·20-s − 0.820·22-s − 0.676·23-s − 0.321·25-s − 0.333·26-s + 0.515·28-s + 0.329·29-s − 0.647·31-s + 0.176·32-s + 0.229·34-s + 0.850·35-s − 0.798·37-s + 0.656·38-s + 0.291·40-s + 1.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.338757652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.338757652\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 7.94T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−7.60676238504556472637118227706, −7.31917973257090861965750791114, −6.19536691732185871690162796208, −5.45073392950490733250629183976, −5.29641223125111383668698626824, −4.40663103515052062651398670718, −3.57571446422407515172446230034, −2.43660903677284458173983256641, −2.14521265789877067684077579955, −0.940644388369713509827820596410,
0.940644388369713509827820596410, 2.14521265789877067684077579955, 2.43660903677284458173983256641, 3.57571446422407515172446230034, 4.40663103515052062651398670718, 5.29641223125111383668698626824, 5.45073392950490733250629183976, 6.19536691732185871690162796208, 7.31917973257090861965750791114, 7.60676238504556472637118227706
