| L(s) = 1 | − 1.78·3-s + 2.65·7-s + 0.174·9-s − 2.07·11-s + 2.29·13-s + 17-s + 0.348·19-s − 4.73·21-s − 0.130·23-s + 5.03·27-s − 0.348·29-s − 1.54·31-s + 3.70·33-s + 3.65·37-s − 4.09·39-s − 5.49·41-s + 4.29·43-s − 6.20·47-s + 0.0708·49-s − 1.78·51-s + 13.8·53-s − 0.621·57-s − 10.4·59-s + 10.9·61-s + 0.463·63-s − 3.86·67-s + 0.232·69-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 1.00·7-s + 0.0580·9-s − 0.626·11-s + 0.637·13-s + 0.242·17-s + 0.0799·19-s − 1.03·21-s − 0.0271·23-s + 0.968·27-s − 0.0647·29-s − 0.278·31-s + 0.644·33-s + 0.600·37-s − 0.655·39-s − 0.858·41-s + 0.655·43-s − 0.905·47-s + 0.0101·49-s − 0.249·51-s + 1.89·53-s − 0.0822·57-s − 1.35·59-s + 1.40·61-s + 0.0583·63-s − 0.471·67-s + 0.0279·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213025385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213025385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 19 | \( 1 - 0.348T + 19T^{2} \) |
| 23 | \( 1 + 0.130T + 23T^{2} \) |
| 29 | \( 1 + 0.348T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 3.89T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−9.356728929317497444248523940218, −8.391341930022184866730851606022, −7.84728737748169287886003547187, −6.83114043923749782941957945059, −5.96490993717110337700123081821, −5.27707790571726129904330509202, −4.65312189245621897469332139138, −3.46134639431238277981898164413, −2.12062671689427858417756962869, −0.813224148095549335930296401439,
0.813224148095549335930296401439, 2.12062671689427858417756962869, 3.46134639431238277981898164413, 4.65312189245621897469332139138, 5.27707790571726129904330509202, 5.96490993717110337700123081821, 6.83114043923749782941957945059, 7.84728737748169287886003547187, 8.391341930022184866730851606022, 9.356728929317497444248523940218
