| L(s) = 1 | + 3-s − 7-s − 2·9-s + 11-s + 4·17-s − 6·19-s − 21-s − 5·23-s − 5·25-s − 5·27-s − 8·29-s − 31-s + 33-s − 5·37-s + 7·41-s − 8·43-s + 7·47-s + 49-s + 4·51-s − 4·53-s − 6·57-s − 10·59-s − 7·61-s + 2·63-s − 7·67-s − 5·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.970·17-s − 1.37·19-s − 0.218·21-s − 1.04·23-s − 25-s − 0.962·27-s − 1.48·29-s − 0.179·31-s + 0.174·33-s − 0.821·37-s + 1.09·41-s − 1.21·43-s + 1.02·47-s + 1/7·49-s + 0.560·51-s − 0.549·53-s − 0.794·57-s − 1.30·59-s − 0.896·61-s + 0.251·63-s − 0.855·67-s − 0.601·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6038378705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6038378705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−14.13717124888849, −13.57001168129901, −13.19804224082205, −12.50069092375638, −12.11560242882897, −11.67009124472290, −10.95043749923964, −10.59650373931535, −9.952443677411732, −9.314433091089838, −9.173046567037947, −8.406993538369610, −7.903119019769136, −7.637524274588731, −6.797659855891618, −6.259444630905487, −5.672373579310352, −5.408013622588147, −4.238249532163244, −4.012114114322355, −3.311560017188356, −2.782298782941160, −1.982502716976410, −1.569741670412083, −0.2280597315766303,
0.2280597315766303, 1.569741670412083, 1.982502716976410, 2.782298782941160, 3.311560017188356, 4.012114114322355, 4.238249532163244, 5.408013622588147, 5.672373579310352, 6.259444630905487, 6.797659855891618, 7.637524274588731, 7.903119019769136, 8.406993538369610, 9.173046567037947, 9.314433091089838, 9.952443677411732, 10.59650373931535, 10.95043749923964, 11.67009124472290, 12.11560242882897, 12.50069092375638, 13.19804224082205, 13.57001168129901, 14.13717124888849
