L(s) = 1 | + (1.11 − 0.868i)2-s + (−0.502 − 0.290i)3-s + (0.491 − 1.93i)4-s + (−0.5 − 0.866i)5-s + (−0.813 + 0.112i)6-s + (−2.63 − 0.253i)7-s + (−1.13 − 2.59i)8-s + (−1.33 − 2.30i)9-s + (−1.31 − 0.532i)10-s + (0.428 − 0.742i)11-s + (−0.809 + 0.832i)12-s + 2.26·13-s + (−3.15 + 2.00i)14-s + 0.580i·15-s + (−3.51 − 1.90i)16-s + (6.65 + 3.84i)17-s + ⋯ |
L(s) = 1 | + (0.789 − 0.614i)2-s + (−0.290 − 0.167i)3-s + (0.245 − 0.969i)4-s + (−0.223 − 0.387i)5-s + (−0.331 + 0.0460i)6-s + (−0.995 − 0.0956i)7-s + (−0.401 − 0.915i)8-s + (−0.443 − 0.768i)9-s + (−0.414 − 0.168i)10-s + (0.129 − 0.223i)11-s + (−0.233 + 0.240i)12-s + 0.627·13-s + (−0.844 + 0.535i)14-s + 0.149i·15-s + (−0.879 − 0.476i)16-s + (1.61 + 0.931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.695600 - 1.32339i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.695600 - 1.32339i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.11 + 0.868i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.63 + 0.253i)T \) |
good | 3 | \( 1 + (0.502 + 0.290i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.428 + 0.742i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 + (-6.65 - 3.84i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.17 + 2.98i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.17 - 1.83i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.76iT - 29T^{2} \) |
| 31 | \( 1 + (4.53 - 7.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.77 + 2.18i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.780iT - 41T^{2} \) |
| 43 | \( 1 - 7.36T + 43T^{2} \) |
| 47 | \( 1 + (0.206 + 0.358i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.0 - 6.36i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.74 + 4.46i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.51 - 7.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.69iT - 71T^{2} \) |
| 73 | \( 1 + (9.52 + 5.49i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.53 + 2.61i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.58iT - 83T^{2} \) |
| 89 | \( 1 + (-5.85 + 3.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.09iT - 97T^{2} \) |
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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
−11.90779489627540265612831816892, −10.77752406131989921658113319977, −9.778484367184732834969882897945, −8.978295153616558363090859843882, −7.40720063452981012963433922985, −6.09986450660699328452333151116, −5.62145663507355151403548882685, −3.91723335348149277820533480574, −3.15653943310334508380563867476, −0.981636164072589594981007250623,
2.85832856730984936374825640602, 3.83030939126698927775116095103, 5.34998030912805882346387230373, 5.99609323746343097185457719493, 7.24090511601816300863745940730, 7.956456916084815892019891843352, 9.333284954633434790732411182736, 10.40168168011171235077313014490, 11.54471925400442088219290366668, 12.18359335683324583130216751941