| L(s) = 1 | + 3·3-s + 5-s − 7-s + 6·9-s + 2·11-s − 7·13-s + 3·15-s + 4·17-s + 19-s − 3·21-s − 3·23-s + 25-s + 9·27-s + 9·29-s − 4·31-s + 6·33-s − 35-s + 3·37-s − 21·39-s + 5·43-s + 6·45-s + 8·47-s + 49-s + 12·51-s + 12·53-s + 2·55-s + 3·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 0.603·11-s − 1.94·13-s + 0.774·15-s + 0.970·17-s + 0.229·19-s − 0.654·21-s − 0.625·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s − 0.718·31-s + 1.04·33-s − 0.169·35-s + 0.493·37-s − 3.36·39-s + 0.762·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s + 1.68·51-s + 1.64·53-s + 0.269·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.930556073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.930556073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 617 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−12.65040229379351, −12.22133593468604, −11.92115735671848, −11.20356017964328, −10.36547215192617, −10.10221113579356, −9.726813571820533, −9.492497829046042, −8.825646931609536, −8.616340740247111, −7.933854631415515, −7.518759173300473, −7.174464678821413, −6.767647528330565, −6.043913038996991, −5.506532961093052, −4.954794258187065, −4.231259810216635, −4.042496917874568, −3.246806132431056, −2.856252331298365, −2.400541661756193, −2.013013633863750, −1.233089940031432, −0.6096767378546461,
0.6096767378546461, 1.233089940031432, 2.013013633863750, 2.400541661756193, 2.856252331298365, 3.246806132431056, 4.042496917874568, 4.231259810216635, 4.954794258187065, 5.506532961093052, 6.043913038996991, 6.767647528330565, 7.174464678821413, 7.518759173300473, 7.933854631415515, 8.616340740247111, 8.825646931609536, 9.492497829046042, 9.726813571820533, 10.10221113579356, 10.36547215192617, 11.20356017964328, 11.92115735671848, 12.22133593468604, 12.65040229379351
