| L(s) = 1 | + 4·11-s − 2i·13-s + 2i·17-s − 4·19-s − 8i·23-s + 6·29-s − 8·31-s − 6i·37-s + 6·41-s − 4i·43-s + 7·49-s + 2i·53-s − 4·59-s − 2·61-s − 4i·67-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554i·13-s + 0.485i·17-s − 0.917·19-s − 1.66i·23-s + 1.11·29-s − 1.43·31-s − 0.986i·37-s + 0.937·41-s − 0.609i·43-s + 49-s + 0.274i·53-s − 0.520·59-s − 0.256·61-s − 0.488i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.741035480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.741035480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.649792764376536655361070287795, −7.68336374031468174486449227595, −6.84837289967856133824018793257, −6.25023235939032560187135724992, −5.51593197042013070059121188915, −4.38693585929860151054920179402, −3.93082385162742754470406013565, −2.79828932804449657071578923926, −1.82982338782789702157272157662, −0.57218158202813785377166073906,
1.14152182841781339007678171368, 2.08960821207701551930732862496, 3.27685745960515832228583320113, 4.04987425194175295140642404569, 4.79624636091922179793487598575, 5.78439418326447439891819911254, 6.49298552746777437529804781851, 7.14769381142733388377539407613, 7.901922215824919972883868166850, 8.880235549524611474629892471213
