| L(s) = 1 | − 4.18i·3-s + 45.6i·5-s − 34.3i·7-s + 63.4·9-s + 103.·11-s + 134.·13-s + 191.·15-s + 240.·17-s + 100. i·19-s − 143.·21-s + 475.·23-s − 1.46e3·25-s − 605. i·27-s − 159. i·29-s − 1.11e3·31-s + ⋯ |
| L(s) = 1 | − 0.465i·3-s + 1.82i·5-s − 0.700i·7-s + 0.783·9-s + 0.852·11-s + 0.798·13-s + 0.850·15-s + 0.831·17-s + 0.279i·19-s − 0.326·21-s + 0.899·23-s − 2.33·25-s − 0.830i·27-s − 0.190i·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.736228618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736228618\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (1.74e3 - 619. i)T \) |
| good | 3 | \( 1 + 4.18iT - 81T^{2} \) |
| 5 | \( 1 - 45.6iT - 625T^{2} \) |
| 7 | \( 1 + 34.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 103.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 134.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 240.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 100. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 475.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 159. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.11e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.45e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.06e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 2.93e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 825.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.24e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.20e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 265.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.86e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.27e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.95e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.35e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 5.35e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.19e3T + 8.85e7T^{2} \) |
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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
−10.12219649774194657587860652177, −9.228636068569551726829677217456, −7.80220217128287532233169887261, −7.18634649190479868758443086776, −6.65491008094079762776522392932, −5.75758416011279959423419560844, −3.98907772119310393949394999488, −3.46443473844876002660266387218, −2.11414086603097433868990836038, −0.944529140039154651847293922508,
0.909445838639947224819395017005, 1.63635963477861341501775179375, 3.48724812679584272498924912235, 4.43273390861300341591555275649, 5.17036835170799713476566609745, 6.01194466262340409974730942520, 7.31103455617284348070201688031, 8.460449484790924532818413821698, 9.033176650653880564185372226065, 9.524334935234550702136163933641
