| L(s) = 1 | − 0.618·5-s − 7-s + 3·11-s − 1.61·13-s + 6.70·17-s + 0.236·19-s + 23-s − 4.61·25-s + 6.23·29-s − 3·31-s + 0.618·35-s + 0.527·37-s − 5.94·41-s + 2.09·43-s + 4.76·47-s + 49-s − 6.32·53-s − 1.85·55-s + 7.38·59-s − 0.854·61-s + 1.00·65-s − 13.5·67-s + 2.38·71-s + 6.23·73-s − 3·77-s + 6.70·79-s − 1.47·83-s + ⋯ |
| L(s) = 1 | − 0.276·5-s − 0.377·7-s + 0.904·11-s − 0.448·13-s + 1.62·17-s + 0.0541·19-s + 0.208·23-s − 0.923·25-s + 1.15·29-s − 0.538·31-s + 0.104·35-s + 0.0867·37-s − 0.928·41-s + 0.318·43-s + 0.694·47-s + 0.142·49-s − 0.868·53-s − 0.250·55-s + 0.961·59-s − 0.109·61-s + 0.124·65-s − 1.65·67-s + 0.282·71-s + 0.729·73-s − 0.341·77-s + 0.754·79-s − 0.161·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840445956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840445956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 0.236T + 19T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 + 0.854T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.019869663907959982211528225933, −7.43763539432361394242654250639, −6.72141782362359993198264871261, −5.98811959682143940937268491733, −5.28604983352227562016277925963, −4.38831192486960473502742275337, −3.60467791695951807820240116994, −2.97372072573913305483302064347, −1.78443794178621808871467510212, −0.73532796445813936415379525243,
0.73532796445813936415379525243, 1.78443794178621808871467510212, 2.97372072573913305483302064347, 3.60467791695951807820240116994, 4.38831192486960473502742275337, 5.28604983352227562016277925963, 5.98811959682143940937268491733, 6.72141782362359993198264871261, 7.43763539432361394242654250639, 8.019869663907959982211528225933
