| L(s) = 1 | − 25·5-s − 98·7-s − 354·11-s + 404·13-s + 654·17-s − 1.79e3·19-s + 1.08e3·23-s + 625·25-s − 5.75e3·29-s − 1.01e4·31-s + 2.45e3·35-s + 5.55e3·37-s − 1.29e4·41-s + 8.96e3·43-s + 5.40e3·47-s − 7.20e3·49-s + 8.21e3·53-s + 8.85e3·55-s − 3.95e3·59-s + 962·61-s − 1.01e4·65-s + 1.79e4·67-s + 5.61e4·71-s − 8.56e4·73-s + 3.46e4·77-s + 2.60e4·79-s + 9.34e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.882·11-s + 0.663·13-s + 0.548·17-s − 1.14·19-s + 0.425·23-s + 1/5·25-s − 1.27·29-s − 1.90·31-s + 0.338·35-s + 0.666·37-s − 1.20·41-s + 0.739·43-s + 0.356·47-s − 3/7·49-s + 0.401·53-s + 0.394·55-s − 0.147·59-s + 0.0331·61-s − 0.296·65-s + 0.488·67-s + 1.32·71-s − 1.88·73-s + 0.666·77-s + 0.469·79-s + 1.48·83-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
| \(L(3)\) |
\(\approx\) |
\(1.035326883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035326883\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 + 354 T + p^{5} T^{2} \) |
| 13 | \( 1 - 404 T + p^{5} T^{2} \) |
| 17 | \( 1 - 654 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1796 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1080 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5754 T + p^{5} T^{2} \) |
| 31 | \( 1 + 10196 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5552 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12960 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8968 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5400 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8214 T + p^{5} T^{2} \) |
| 59 | \( 1 + 3954 T + p^{5} T^{2} \) |
| 61 | \( 1 - 962 T + p^{5} T^{2} \) |
| 67 | \( 1 - 4 p^{2} T + p^{5} T^{2} \) |
| 71 | \( 1 - 56148 T + p^{5} T^{2} \) |
| 73 | \( 1 + 85690 T + p^{5} T^{2} \) |
| 79 | \( 1 - 26044 T + p^{5} T^{2} \) |
| 83 | \( 1 - 93468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73428 T + p^{5} T^{2} \) |
| 97 | \( 1 - 128978 T + p^{5} T^{2} \) |
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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
−9.605393188937412394730917029610, −8.804674599891702530139431978456, −7.891409377277992601968383424166, −7.09510908612980198958599773381, −6.08893501332529822774986737848, −5.23376464720659304359442359503, −3.97068358789437221793261809497, −3.20897641824935571450041187587, −1.96649012151838268757691724523, −0.45684057416041651322849728438,
0.45684057416041651322849728438, 1.96649012151838268757691724523, 3.20897641824935571450041187587, 3.97068358789437221793261809497, 5.23376464720659304359442359503, 6.08893501332529822774986737848, 7.09510908612980198958599773381, 7.891409377277992601968383424166, 8.804674599891702530139431978456, 9.605393188937412394730917029610
