| L(s) = 1 | − 100·5-s − 343·7-s − 2.77e3·11-s − 3.29e3·13-s − 5.90e3·17-s + 6.64e3·19-s − 1.98e3·23-s − 6.81e4·25-s + 2.08e5·29-s − 1.17e5·31-s + 3.43e4·35-s − 3.35e5·37-s + 2.65e5·41-s − 9.32e4·43-s + 6.57e5·47-s + 1.17e5·49-s + 6.08e5·53-s + 2.77e5·55-s + 5.36e5·59-s − 1.79e6·61-s + 3.29e5·65-s + 2.12e6·67-s + 1.19e6·71-s + 1.05e6·73-s + 9.51e5·77-s + 9.98e5·79-s − 3.89e6·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 0.377·7-s − 0.628·11-s − 0.415·13-s − 0.291·17-s + 0.222·19-s − 0.0339·23-s − 0.871·25-s + 1.58·29-s − 0.710·31-s + 0.135·35-s − 1.08·37-s + 0.601·41-s − 0.178·43-s + 0.923·47-s + 1/7·49-s + 0.561·53-s + 0.224·55-s + 0.339·59-s − 1.01·61-s + 0.148·65-s + 0.862·67-s + 0.394·71-s + 0.317·73-s + 0.237·77-s + 0.227·79-s − 0.748·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.317166856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317166856\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 4 p^{2} T + p^{7} T^{2} \) |
| 11 | \( 1 + 2774 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3294 T + p^{7} T^{2} \) |
| 17 | \( 1 + 5900 T + p^{7} T^{2} \) |
| 19 | \( 1 - 6644 T + p^{7} T^{2} \) |
| 23 | \( 1 + 1982 T + p^{7} T^{2} \) |
| 29 | \( 1 - 208106 T + p^{7} T^{2} \) |
| 31 | \( 1 + 117792 T + p^{7} T^{2} \) |
| 37 | \( 1 + 335686 T + p^{7} T^{2} \) |
| 41 | \( 1 - 265488 T + p^{7} T^{2} \) |
| 43 | \( 1 + 93292 T + p^{7} T^{2} \) |
| 47 | \( 1 - 657516 T + p^{7} T^{2} \) |
| 53 | \( 1 - 608718 T + p^{7} T^{2} \) |
| 59 | \( 1 - 536120 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1797090 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2123176 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1191214 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1056430 T + p^{7} T^{2} \) |
| 79 | \( 1 - 998484 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3898004 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4622352 T + p^{7} T^{2} \) |
| 97 | \( 1 - 15287710 T + p^{7} T^{2} \) |
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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
−10.72544911227137829316975528226, −9.917145854374944407909832518393, −8.844270400208669519473647219817, −7.82655258821422345976416271711, −6.91545686590752346873852887979, −5.72336211121508169178957754832, −4.59847020299536631694611403790, −3.38301691945411161704719336835, −2.19933198981379827249600908107, −0.56249349651547869118352319677,
0.56249349651547869118352319677, 2.19933198981379827249600908107, 3.38301691945411161704719336835, 4.59847020299536631694611403790, 5.72336211121508169178957754832, 6.91545686590752346873852887979, 7.82655258821422345976416271711, 8.844270400208669519473647219817, 9.917145854374944407909832518393, 10.72544911227137829316975528226
