| L(s) = 1 | − 25·5-s − 128·7-s − 308·11-s − 1.05e3·13-s − 1.58e3·17-s − 2.30e3·19-s + 2.65e3·23-s + 625·25-s − 1.19e3·29-s − 9.52e3·31-s + 3.20e3·35-s + 4.47e3·37-s + 6.19e3·41-s + 6.33e3·43-s + 1.49e4·47-s − 423·49-s − 3.83e4·53-s + 7.70e3·55-s + 1.15e4·59-s − 4.83e4·61-s + 2.64e4·65-s − 5.69e4·67-s + 4.48e4·71-s − 1.94e4·73-s + 3.94e4·77-s + 7.73e4·79-s + 4.03e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.987·7-s − 0.767·11-s − 1.73·13-s − 1.33·17-s − 1.46·19-s + 1.04·23-s + 1/5·25-s − 0.264·29-s − 1.77·31-s + 0.441·35-s + 0.536·37-s + 0.575·41-s + 0.522·43-s + 0.985·47-s − 0.0251·49-s − 1.87·53-s + 0.343·55-s + 0.432·59-s − 1.66·61-s + 0.776·65-s − 1.55·67-s + 1.05·71-s − 0.427·73-s + 0.757·77-s + 1.39·79-s + 0.643·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1806094249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1806094249\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 128 T + p^{5} T^{2} \) |
| 11 | \( 1 + 28 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 1058 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1586 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2308 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2656 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1198 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9520 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4470 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6198 T + p^{5} T^{2} \) |
| 43 | \( 1 - 6332 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14920 T + p^{5} T^{2} \) |
| 53 | \( 1 + 38310 T + p^{5} T^{2} \) |
| 59 | \( 1 - 196 p T + p^{5} T^{2} \) |
| 61 | \( 1 + 48338 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56972 T + p^{5} T^{2} \) |
| 71 | \( 1 - 44856 T + p^{5} T^{2} \) |
| 73 | \( 1 + 19446 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77328 T + p^{5} T^{2} \) |
| 83 | \( 1 - 40364 T + p^{5} T^{2} \) |
| 89 | \( 1 + 35706 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97022 T + p^{5} T^{2} \) |
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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
−9.494271440362645045520837981128, −8.992085263321549188833063630618, −7.76160938529579231932300514314, −7.09175014168280524640705495211, −6.23123778774458421919448371159, −5.02583476886649997322093907449, −4.21700048997774970921520994376, −2.95523025915351271566441749048, −2.15892814071712719292215053619, −0.18210061773419486465587423740,
0.18210061773419486465587423740, 2.15892814071712719292215053619, 2.95523025915351271566441749048, 4.21700048997774970921520994376, 5.02583476886649997322093907449, 6.23123778774458421919448371159, 7.09175014168280524640705495211, 7.76160938529579231932300514314, 8.992085263321549188833063630618, 9.494271440362645045520837981128
