| L(s) = 1 | − 2.45·3-s − 2.10·5-s + 3.05·9-s − 2.99·11-s + 0.922·13-s + 5.16·15-s + 6.22·17-s + 6.64·19-s − 6.34·23-s − 0.583·25-s − 0.124·27-s − 1.64·29-s + 3.00·31-s + 7.36·33-s + 9.04·37-s − 2.26·39-s − 41-s + 7.65·43-s − 6.41·45-s − 6.66·47-s − 15.3·51-s − 7.74·53-s + 6.29·55-s − 16.3·57-s − 13.3·59-s + 9.83·61-s − 1.93·65-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 0.939·5-s + 1.01·9-s − 0.903·11-s + 0.255·13-s + 1.33·15-s + 1.50·17-s + 1.52·19-s − 1.32·23-s − 0.116·25-s − 0.0239·27-s − 0.304·29-s + 0.539·31-s + 1.28·33-s + 1.48·37-s − 0.363·39-s − 0.156·41-s + 1.16·43-s − 0.955·45-s − 0.972·47-s − 2.14·51-s − 1.06·53-s + 0.848·55-s − 2.16·57-s − 1.73·59-s + 1.25·61-s − 0.240·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6744494598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6744494598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 - 0.922T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 9.83T + 61T^{2} \) |
| 67 | \( 1 + 8.26T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 - 0.193T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.71494894717716193707103608107, −7.33163278334833498592953530659, −6.07588396584652573299861260255, −5.91220741536010921072815672515, −5.08304931487043183270744191207, −4.48277542579392067978715939541, −3.58518191129967595136194383016, −2.84918406255436038150798141174, −1.38320120354664723478875436198, −0.47504354409650531923943355699,
0.47504354409650531923943355699, 1.38320120354664723478875436198, 2.84918406255436038150798141174, 3.58518191129967595136194383016, 4.48277542579392067978715939541, 5.08304931487043183270744191207, 5.91220741536010921072815672515, 6.07588396584652573299861260255, 7.33163278334833498592953530659, 7.71494894717716193707103608107
