| L(s) = 1 | + 96·5-s + 49·7-s + 720·11-s + 572·13-s − 1.25e3·17-s − 94·19-s − 96·23-s + 6.09e3·25-s + 4.37e3·29-s − 6.24e3·31-s + 4.70e3·35-s − 1.07e4·37-s − 1.20e4·41-s − 9.16e3·43-s + 2.58e4·47-s + 2.40e3·49-s − 1.01e3·53-s + 6.91e4·55-s − 1.24e3·59-s + 7.59e3·61-s + 5.49e4·65-s + 4.11e4·67-s + 3.76e4·71-s − 1.34e4·73-s + 3.52e4·77-s + 6.24e3·79-s + 2.52e4·83-s + ⋯ |
| L(s) = 1 | + 1.71·5-s + 0.377·7-s + 1.79·11-s + 0.938·13-s − 1.05·17-s − 0.0597·19-s − 0.0378·23-s + 1.94·25-s + 0.965·29-s − 1.16·31-s + 0.649·35-s − 1.29·37-s − 1.11·41-s − 0.755·43-s + 1.70·47-s + 1/7·49-s − 0.0495·53-s + 3.08·55-s − 0.0464·59-s + 0.261·61-s + 1.61·65-s + 1.11·67-s + 0.885·71-s − 0.295·73-s + 0.678·77-s + 0.112·79-s + 0.402·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.417232203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417232203\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 96 T + p^{5} T^{2} \) |
| 11 | \( 1 - 720 T + p^{5} T^{2} \) |
| 13 | \( 1 - 44 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1254 T + p^{5} T^{2} \) |
| 19 | \( 1 + 94 T + p^{5} T^{2} \) |
| 23 | \( 1 + 96 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4374 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6244 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12006 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9160 T + p^{5} T^{2} \) |
| 47 | \( 1 - 25836 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1014 T + p^{5} T^{2} \) |
| 59 | \( 1 + 1242 T + p^{5} T^{2} \) |
| 61 | \( 1 - 7592 T + p^{5} T^{2} \) |
| 67 | \( 1 - 41132 T + p^{5} T^{2} \) |
| 71 | \( 1 - 37632 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13438 T + p^{5} T^{2} \) |
| 79 | \( 1 - 6248 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25254 T + p^{5} T^{2} \) |
| 89 | \( 1 - 45126 T + p^{5} T^{2} \) |
| 97 | \( 1 - 107222 T + p^{5} T^{2} \) |
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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
−11.07997769004327497873864175886, −10.19905114445934005235328822324, −9.113586937523096138048897295155, −8.708036719053210425503338138769, −6.80923408949690372420803922762, −6.25776464124503790928344988874, −5.13878585087194986800969731221, −3.76771266103547017919565726086, −2.07592691154418043550096265874, −1.24359954126291028944787993055,
1.24359954126291028944787993055, 2.07592691154418043550096265874, 3.76771266103547017919565726086, 5.13878585087194986800969731221, 6.25776464124503790928344988874, 6.80923408949690372420803922762, 8.708036719053210425503338138769, 9.113586937523096138048897295155, 10.19905114445934005235328822324, 11.07997769004327497873864175886
