| L(s) = 1 | + 64·5-s − 49·7-s − 54·11-s + 738·13-s + 848·17-s + 1.60e3·19-s − 3.67e3·23-s + 971·25-s + 4.33e3·29-s + 4.76e3·31-s − 3.13e3·35-s − 2.09e3·37-s + 6.11e3·41-s − 7.91e3·43-s + 6.57e3·47-s + 2.40e3·49-s + 7.89e3·53-s − 3.45e3·55-s − 4.16e4·59-s − 2.65e4·61-s + 4.72e4·65-s + 4.17e4·67-s + 8.35e4·71-s − 4.23e4·73-s + 2.64e3·77-s − 508·79-s − 8.36e3·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s − 0.377·7-s − 0.134·11-s + 1.21·13-s + 0.711·17-s + 1.01·19-s − 1.44·23-s + 0.310·25-s + 0.956·29-s + 0.889·31-s − 0.432·35-s − 0.251·37-s + 0.568·41-s − 0.652·43-s + 0.433·47-s + 1/7·49-s + 0.386·53-s − 0.154·55-s − 1.55·59-s − 0.914·61-s + 1.38·65-s + 1.13·67-s + 1.96·71-s − 0.929·73-s + 0.0508·77-s − 0.00915·79-s − 0.133·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.206560284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.206560284\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 64 T + p^{5} T^{2} \) |
| 11 | \( 1 + 54 T + p^{5} T^{2} \) |
| 13 | \( 1 - 738 T + p^{5} T^{2} \) |
| 17 | \( 1 - 848 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1604 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3670 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4330 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4760 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2094 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6116 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7916 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6572 T + p^{5} T^{2} \) |
| 53 | \( 1 - 7894 T + p^{5} T^{2} \) |
| 59 | \( 1 + 41664 T + p^{5} T^{2} \) |
| 61 | \( 1 + 26570 T + p^{5} T^{2} \) |
| 67 | \( 1 - 41736 T + p^{5} T^{2} \) |
| 71 | \( 1 - 83574 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42314 T + p^{5} T^{2} \) |
| 79 | \( 1 + 508 T + p^{5} T^{2} \) |
| 83 | \( 1 + 8364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 49220 T + p^{5} T^{2} \) |
| 97 | \( 1 - 159670 T + p^{5} T^{2} \) |
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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
−9.396644029239203117810905072874, −8.445667897096370655432241008546, −7.62708042673702411195328832382, −6.37158289074134551924054740732, −5.99175259430479989513110929299, −5.07375505568830005126777197290, −3.81553028674482924548278676237, −2.86872714420579214548765220803, −1.75221008797459503710551464772, −0.812830223911396921041694744465,
0.812830223911396921041694744465, 1.75221008797459503710551464772, 2.86872714420579214548765220803, 3.81553028674482924548278676237, 5.07375505568830005126777197290, 5.99175259430479989513110929299, 6.37158289074134551924054740732, 7.62708042673702411195328832382, 8.445667897096370655432241008546, 9.396644029239203117810905072874
