| L(s) = 1 | − 27·3-s + 100·5-s − 343·7-s + 729·9-s + 2.77e3·11-s − 3.29e3·13-s − 2.70e3·15-s + 5.90e3·17-s + 6.64e3·19-s + 9.26e3·21-s + 1.98e3·23-s − 6.81e4·25-s − 1.96e4·27-s − 2.08e5·29-s − 1.17e5·31-s − 7.48e4·33-s − 3.43e4·35-s − 3.35e5·37-s + 8.89e4·39-s − 2.65e5·41-s − 9.32e4·43-s + 7.29e4·45-s − 6.57e5·47-s + 1.17e5·49-s − 1.59e5·51-s − 6.08e5·53-s + 2.77e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.357·5-s − 0.377·7-s + 1/3·9-s + 0.628·11-s − 0.415·13-s − 0.206·15-s + 0.291·17-s + 0.222·19-s + 0.218·21-s + 0.0339·23-s − 0.871·25-s − 0.192·27-s − 1.58·29-s − 0.710·31-s − 0.362·33-s − 0.135·35-s − 1.08·37-s + 0.240·39-s − 0.601·41-s − 0.178·43-s + 0.119·45-s − 0.923·47-s + 1/7·49-s − 0.168·51-s − 0.561·53-s + 0.224·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 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
−12.26327752688485113692119052833, −11.28963543910156009299305564943, −10.05912654876854082151654391176, −9.168632393704708287271195018394, −7.51876199311847863803506807154, −6.33469311027517854523544770523, −5.21606738572037131193301105895, −3.62403721098224109192558892109, −1.72781302756464489008368950815, 0,
1.72781302756464489008368950815, 3.62403721098224109192558892109, 5.21606738572037131193301105895, 6.33469311027517854523544770523, 7.51876199311847863803506807154, 9.168632393704708287271195018394, 10.05912654876854082151654391176, 11.28963543910156009299305564943, 12.26327752688485113692119052833
