| L(s) = 1 | + 0.513i·3-s − 1.12·7-s + 2.73·9-s + 3.99i·11-s + 1.12i·13-s + 2.93·17-s + 3.24i·19-s − 0.579i·21-s + 5.39·23-s + 2.94i·27-s + 7.35i·29-s − 7.19·31-s − 2.05·33-s − 10.6i·37-s − 0.576·39-s + ⋯ |
| L(s) = 1 | + 0.296i·3-s − 0.426·7-s + 0.912·9-s + 1.20i·11-s + 0.311i·13-s + 0.713·17-s + 0.743i·19-s − 0.126i·21-s + 1.12·23-s + 0.566i·27-s + 1.36i·29-s − 1.29·31-s − 0.357·33-s − 1.74i·37-s − 0.0923·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694269392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694269392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.513iT - 3T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 - 3.99iT - 11T^{2} \) |
| 13 | \( 1 - 1.12iT - 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 3.24iT - 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 - 7.35iT - 29T^{2} \) |
| 31 | \( 1 + 7.19T + 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 + 6.26iT - 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 1.64iT - 53T^{2} \) |
| 59 | \( 1 - 5.55iT - 59T^{2} \) |
| 61 | \( 1 + 6.65iT - 61T^{2} \) |
| 67 | \( 1 - 6.79iT - 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 6.97T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.920143174993065417690150633372, −7.70083980673035652881735308931, −7.21074619455809283939682021105, −6.66613837762899650284157075936, −5.52778571415985670818300735440, −4.95006497173334741326292456849, −4.00053746121733799513036990331, −3.44529258568367894657878184796, −2.17531639147917541292883055200, −1.28235258219503554861199290963,
0.52312397280325900042471801986, 1.50838601524520330227909747796, 2.84414517113314014830536619361, 3.42634486215447078799071590672, 4.46213086393710615552931071448, 5.28464014146204934558139208633, 6.15415618446642476157125486598, 6.71394432380531236558854849892, 7.54857410115886739008505352788, 8.133267047091115264411502045950
