L(s) = 1 | + (1.83 − 1.27i)5-s + 0.920·7-s + 4.88·11-s − 0.822i·13-s − 5.93i·17-s + 7.43i·19-s + (−2.12 + 4.29i)23-s + (1.75 − 4.68i)25-s − 2.63i·29-s + 5.24·31-s + (1.69 − 1.17i)35-s + 7.23·37-s + 1.93i·41-s − 1.42·43-s + 13.2·47-s + ⋯ |
L(s) = 1 | + (0.821 − 0.569i)5-s + 0.347·7-s + 1.47·11-s − 0.228i·13-s − 1.43i·17-s + 1.70i·19-s + (−0.443 + 0.896i)23-s + (0.350 − 0.936i)25-s − 0.488i·29-s + 0.941·31-s + (0.285 − 0.198i)35-s + 1.18·37-s + 0.301i·41-s − 0.217·43-s + 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.737625419\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.737625419\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.83 + 1.27i)T \) |
| 23 | \( 1 + (2.12 - 4.29i)T \) |
good | 7 | \( 1 - 0.920T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 0.822iT - 13T^{2} \) |
| 17 | \( 1 + 5.93iT - 17T^{2} \) |
| 19 | \( 1 - 7.43iT - 19T^{2} \) |
| 29 | \( 1 + 2.63iT - 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 - 1.93iT - 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.0569iT - 53T^{2} \) |
| 59 | \( 1 - 2.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.95iT - 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 8.61iT - 71T^{2} \) |
| 73 | \( 1 - 3.83iT - 73T^{2} \) |
| 79 | \( 1 + 5.42iT - 79T^{2} \) |
| 83 | \( 1 + 1.52iT - 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.354219180102450754319821989032, −7.74814277949993556653051794431, −6.81702081118688174801679326930, −6.02423435218676488651472926058, −5.54056789960198574763621351850, −4.55951126646250756578089691910, −3.92026852459742003408516573035, −2.78155947232956653969468365356, −1.71084829914973424237543259582, −0.966647099727407134971856811196,
1.08536163466488168867513590405, 2.03709836792877624274048113875, 2.88606468669575940410042172334, 3.99333120078959157660777337827, 4.60376522450170138784665292797, 5.67304016109160303779526789752, 6.47492683578729388847384645083, 6.69508825839481337554503824451, 7.70240890275167875469070480088, 8.680316249219364867725362582842