| L(s) = 1 | + 2·7-s − 11-s + 5·13-s − 2·17-s + 8·23-s − 5·25-s + 5·29-s + 2·37-s + 2·41-s − 11·43-s − 9·47-s − 3·49-s + 2·53-s + 14·59-s + 7·61-s + 2·67-s − 3·71-s − 4·73-s − 2·77-s + 17·83-s + 7·89-s + 10·91-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 1.38·13-s − 0.485·17-s + 1.66·23-s − 25-s + 0.928·29-s + 0.328·37-s + 0.312·41-s − 1.67·43-s − 1.31·47-s − 3/7·49-s + 0.274·53-s + 1.82·59-s + 0.896·61-s + 0.244·67-s − 0.356·71-s − 0.468·73-s − 0.227·77-s + 1.86·83-s + 0.741·89-s + 1.04·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.385559769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.385559769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.00114583820283, −12.17930707650623, −11.66773599448276, −11.32525181545711, −11.01781848154754, −10.53804978503363, −9.958205314950929, −9.569696190481849, −8.879678812698532, −8.477626567875344, −8.235736171838056, −7.700396590716248, −7.049524490501907, −6.562197193492925, −6.271266187974615, −5.430261921715843, −5.191914282047039, −4.610943065679271, −4.082074988307346, −3.479136526250693, −3.031147296661547, −2.303732683565207, −1.715196605999622, −1.158657297381917, −0.5272634057800945,
0.5272634057800945, 1.158657297381917, 1.715196605999622, 2.303732683565207, 3.031147296661547, 3.479136526250693, 4.082074988307346, 4.610943065679271, 5.191914282047039, 5.430261921715843, 6.271266187974615, 6.562197193492925, 7.049524490501907, 7.700396590716248, 8.235736171838056, 8.477626567875344, 8.879678812698532, 9.569696190481849, 9.958205314950929, 10.53804978503363, 11.01781848154754, 11.32525181545711, 11.66773599448276, 12.17930707650623, 13.00114583820283
