| L(s) = 1 | + 2·5-s + 11-s + 2·17-s − 6·19-s − 6·23-s − 25-s + 2·29-s + 2·37-s − 2·41-s + 6·43-s + 6·47-s + 2·53-s + 2·55-s + 8·61-s + 12·67-s + 6·71-s − 6·73-s − 12·79-s + 12·83-s + 4·85-s − 8·89-s − 12·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.485·17-s − 1.37·19-s − 1.25·23-s − 1/5·25-s + 0.371·29-s + 0.328·37-s − 0.312·41-s + 0.914·43-s + 0.875·47-s + 0.274·53-s + 0.269·55-s + 1.02·61-s + 1.46·67-s + 0.712·71-s − 0.702·73-s − 1.35·79-s + 1.31·83-s + 0.433·85-s − 0.847·89-s − 1.23·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.535921282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.535921282\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.64780690211932, −12.24350427214340, −11.83503314060182, −11.21056610201396, −10.82574807045743, −10.23554421564249, −9.947316615585163, −9.579421798022420, −8.945418513990457, −8.581428399305829, −8.056210685968332, −7.616492102720255, −6.972582056573236, −6.419869770566772, −6.171137194310968, −5.572179584104233, −5.258221496336364, −4.460157923613965, −3.994943069030499, −3.666017710871845, −2.663792612504287, −2.387778139895502, −1.804237216754266, −1.206558528492456, −0.4192610693099345,
0.4192610693099345, 1.206558528492456, 1.804237216754266, 2.387778139895502, 2.663792612504287, 3.666017710871845, 3.994943069030499, 4.460157923613965, 5.258221496336364, 5.572179584104233, 6.171137194310968, 6.419869770566772, 6.972582056573236, 7.616492102720255, 8.056210685968332, 8.581428399305829, 8.945418513990457, 9.579421798022420, 9.947316615585163, 10.23554421564249, 10.82574807045743, 11.21056610201396, 11.83503314060182, 12.24350427214340, 12.64780690211932
