L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.03 − 0.598i)5-s + (2.52 − 0.782i)7-s − 0.999·8-s + (−1.03 + 0.598i)10-s + (2.65 + 1.98i)11-s − 3.26i·13-s + (0.586 − 2.57i)14-s + (−0.5 + 0.866i)16-s + (−2.21 − 3.82i)17-s + (6.73 + 3.88i)19-s + 1.19i·20-s + (3.04 − 1.30i)22-s + (5.94 + 3.43i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.463 − 0.267i)5-s + (0.955 − 0.295i)7-s − 0.353·8-s + (−0.327 + 0.189i)10-s + (0.800 + 0.599i)11-s − 0.904i·13-s + (0.156 − 0.689i)14-s + (−0.125 + 0.216i)16-s + (−0.536 − 0.928i)17-s + (1.54 + 0.892i)19-s + 0.267i·20-s + (0.650 − 0.277i)22-s + (1.24 + 0.715i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.038008286\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.038008286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.52 + 0.782i)T \) |
| 11 | \( 1 + (-2.65 - 1.98i)T \) |
good | 5 | \( 1 + (1.03 + 0.598i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 3.26iT - 13T^{2} \) |
| 17 | \( 1 + (2.21 + 3.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.73 - 3.88i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.94 - 3.43i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.85T + 29T^{2} \) |
| 31 | \( 1 + (0.103 + 0.179i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 6.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 + 8.00iT - 43T^{2} \) |
| 47 | \( 1 + (0.355 + 0.205i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.02 + 2.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.66 + 2.11i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.94 + 5.74i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.61 - 9.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.80iT - 71T^{2} \) |
| 73 | \( 1 + (-0.195 + 0.113i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.00 - 1.15i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + (11.4 + 6.58i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.458028022796846122646457828465, −8.664689932406024283304570951803, −7.59289098476145761707427503613, −7.18537949995980182998428493452, −5.65781524125370388978111440735, −5.06889331395286321985040832880, −4.12795810390758446173673378550, −3.34959458164173538944058361798, −1.97146024929734774922672923924, −0.846533791035509974016751478862,
1.40980630962608792275126749288, 2.93665410669562245683150318346, 3.97160451012187332559049897555, 4.75605905517622160629359772631, 5.65011555765338631616681544212, 6.61435878621443914565159642674, 7.28634555598653700980191999854, 8.101939018512844972705041688330, 8.934599899862556495752666894714, 9.416315890517729646687841703400