| L(s) = 1 | + 26·5-s + 1.05e3·7-s − 6.41e3·11-s + 5.20e3·13-s + 6.23e3·17-s + 4.14e4·19-s + 2.94e4·23-s − 7.74e4·25-s + 2.10e5·29-s + 1.85e5·31-s + 2.74e4·35-s + 5.07e5·37-s − 3.60e5·41-s + 6.20e5·43-s + 8.47e5·47-s + 2.91e5·49-s − 1.42e6·53-s − 1.66e5·55-s + 2.54e6·59-s − 7.06e5·61-s + 1.35e5·65-s − 2.41e6·67-s − 2.65e5·71-s − 5.79e6·73-s − 6.77e6·77-s + 2.95e6·79-s − 3.46e6·83-s + ⋯ |
| L(s) = 1 | + 0.0930·5-s + 1.16·7-s − 1.45·11-s + 0.657·13-s + 0.307·17-s + 1.38·19-s + 0.504·23-s − 0.991·25-s + 1.60·29-s + 1.11·31-s + 0.108·35-s + 1.64·37-s − 0.815·41-s + 1.18·43-s + 1.19·47-s + 0.354·49-s − 1.31·53-s − 0.135·55-s + 1.61·59-s − 0.398·61-s + 0.0611·65-s − 0.982·67-s − 0.0881·71-s − 1.74·73-s − 1.69·77-s + 0.674·79-s − 0.664·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.210233427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210233427\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 26 T + p^{7} T^{2} \) |
| 7 | \( 1 - 1056 T + p^{7} T^{2} \) |
| 11 | \( 1 + 6412 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5206 T + p^{7} T^{2} \) |
| 17 | \( 1 - 6238 T + p^{7} T^{2} \) |
| 19 | \( 1 - 41492 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29432 T + p^{7} T^{2} \) |
| 29 | \( 1 - 210498 T + p^{7} T^{2} \) |
| 31 | \( 1 - 185240 T + p^{7} T^{2} \) |
| 37 | \( 1 - 507630 T + p^{7} T^{2} \) |
| 41 | \( 1 + 360042 T + p^{7} T^{2} \) |
| 43 | \( 1 - 620044 T + p^{7} T^{2} \) |
| 47 | \( 1 - 847680 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1423750 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2548724 T + p^{7} T^{2} \) |
| 61 | \( 1 + 706058 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2418796 T + p^{7} T^{2} \) |
| 71 | \( 1 + 265976 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5791238 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2955688 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3462932 T + p^{7} T^{2} \) |
| 89 | \( 1 - 2211126 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15594814 T + p^{7} T^{2} \) |
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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
−13.35995821846221478854343908604, −11.96845050790699125363837039241, −10.99416781275053200170176190645, −9.919673373035440319231734896658, −8.357475860883198387031992486062, −7.56817876632869140362423006323, −5.75926119734725558158135432958, −4.64649807253348555979505098489, −2.76058037610974394286710588709, −1.07269955099675561682148079663,
1.07269955099675561682148079663, 2.76058037610974394286710588709, 4.64649807253348555979505098489, 5.75926119734725558158135432958, 7.56817876632869140362423006323, 8.357475860883198387031992486062, 9.919673373035440319231734896658, 10.99416781275053200170176190645, 11.96845050790699125363837039241, 13.35995821846221478854343908604
