| L(s) = 1 | − 2.51i·2-s − 4.34·4-s + (2.08 + 0.814i)5-s − 4.08i·7-s + 5.91i·8-s + (2.05 − 5.24i)10-s + 1.07·11-s + 1.43i·13-s − 10.2·14-s + 6.20·16-s − 6.88i·17-s + 7.42·19-s + (−9.05 − 3.54i)20-s − 2.71i·22-s − 1.01i·23-s + ⋯ |
| L(s) = 1 | − 1.78i·2-s − 2.17·4-s + (0.931 + 0.364i)5-s − 1.54i·7-s + 2.09i·8-s + (0.649 − 1.65i)10-s + 0.325·11-s + 0.397i·13-s − 2.74·14-s + 1.55·16-s − 1.67i·17-s + 1.70·19-s + (−2.02 − 0.791i)20-s − 0.579i·22-s − 0.212i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620458967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620458967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.08 - 0.814i)T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51iT - 2T^{2} \) |
| 7 | \( 1 + 4.08iT - 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 1.43iT - 13T^{2} \) |
| 17 | \( 1 + 6.88iT - 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 1.01iT - 23T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 6.88iT - 47T^{2} \) |
| 53 | \( 1 + 7.61iT - 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 + 0.265T + 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 3.54iT - 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 0.615iT - 83T^{2} \) |
| 89 | \( 1 - 2.11T + 89T^{2} \) |
| 97 | \( 1 - 1.52iT - 97T^{2} \) |
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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
−9.596797303937147160553066960929, −8.991000517511363714967487429663, −7.48429613240855605811888393432, −6.92165489139613011561750555032, −5.46167964996784014629328450771, −4.62208312084623053947626986983, −3.63504052677711535567047755255, −2.89230156399119772700036415544, −1.70305422072896613229485385881, −0.72472375149696487557192936242,
1.64969312778584402888974133589, 3.23473275191521161443641162536, 4.69423810541791626909407414317, 5.54519709627304857263803364799, 5.83947800651882888836642908165, 6.59394642647028422326211969302, 7.69458349714275299455608582159, 8.434959191639255383241305622265, 9.087484710433083623701253389665, 9.531520998684073690100826952594
