| L(s) = 1 | + 1.19·3-s + 3.34i·5-s + i·7-s − 1.57·9-s − i·11-s + (1.17 − 3.40i)13-s + 3.98i·15-s − 6.04·17-s + 2.04i·19-s + 1.19i·21-s − 2.12·23-s − 6.16·25-s − 5.45·27-s − 6.58·29-s − 11.0i·31-s + ⋯ |
| L(s) = 1 | + 0.688·3-s + 1.49i·5-s + 0.377i·7-s − 0.525·9-s − 0.301i·11-s + (0.325 − 0.945i)13-s + 1.02i·15-s − 1.46·17-s + 0.468i·19-s + 0.260i·21-s − 0.443·23-s − 1.23·25-s − 1.05·27-s − 1.22·29-s − 1.98i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4987035110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4987035110\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.17 + 3.40i)T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 3.34iT - 5T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 - 2.04iT - 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 11.0iT - 31T^{2} \) |
| 37 | \( 1 + 8.33iT - 37T^{2} \) |
| 41 | \( 1 + 9.38iT - 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 - 4.19iT - 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 + 0.541iT - 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 3.32iT - 67T^{2} \) |
| 71 | \( 1 - 1.30iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 - 2.01iT - 83T^{2} \) |
| 89 | \( 1 - 14.4iT - 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−8.042808246497242619955881205783, −7.67586499766795315491435900965, −6.80146415757148972655046072251, −5.92054826521433595419474893489, −5.61618581937212158131753987582, −3.99889144885085604413685633290, −3.55766966169518493775817320122, −2.44640593792819744839159144406, −2.28974366764664500279955211999, −0.12005620967287443810154578860,
1.33572298027041192219998688973, 2.14040010570541630535831481432, 3.27969384796963750820275516955, 4.28959441169296030605885700434, 4.69157918039373678318268692624, 5.59344276129037015203400417774, 6.56073093951938403777410348204, 7.25774202519373755912028082571, 8.344550926717298338010037230205, 8.567086285665843450961645005409
