| L(s) = 1 | + (0.766 + 0.642i)2-s + (−2.46 + 0.898i)3-s + (0.173 + 0.984i)4-s + (0.624 − 3.54i)5-s + (−2.46 − 0.898i)6-s + (2.28 + 3.96i)7-s + (−0.500 + 0.866i)8-s + (2.98 − 2.50i)9-s + (2.75 − 2.31i)10-s + (0.5 − 0.866i)11-s + (−1.31 − 2.27i)12-s + (−0.219 − 0.0798i)13-s + (−0.795 + 4.51i)14-s + (1.64 + 9.30i)15-s + (−0.939 + 0.342i)16-s + (5.19 + 4.35i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−1.42 + 0.518i)3-s + (0.0868 + 0.492i)4-s + (0.279 − 1.58i)5-s + (−1.00 − 0.366i)6-s + (0.865 + 1.49i)7-s + (−0.176 + 0.306i)8-s + (0.994 − 0.834i)9-s + (0.871 − 0.731i)10-s + (0.150 − 0.261i)11-s + (−0.378 − 0.656i)12-s + (−0.0608 − 0.0221i)13-s + (−0.212 + 1.20i)14-s + (0.423 + 2.40i)15-s + (−0.234 + 0.0855i)16-s + (1.25 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968209 + 0.866378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968209 + 0.866378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (1.99 - 3.87i)T \) |
| good | 3 | \( 1 + (2.46 - 0.898i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.624 + 3.54i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.28 - 3.96i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (0.219 + 0.0798i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.19 - 4.35i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.29 - 7.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.81 - 1.52i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.154 - 0.267i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.954T + 37T^{2} \) |
| 41 | \( 1 + (-8.73 + 3.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 8.20i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.10 + 1.76i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.13 - 6.41i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.29 - 1.92i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.21 + 6.90i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.96 - 5.00i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.53 + 14.3i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-15.0 + 5.49i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (9.56 - 3.48i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.819 + 1.42i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.50 + 0.912i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (6.25 + 5.24i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.74954801609673732116103186911, −10.74566603444429881715419626286, −9.489324292954713316966780717672, −8.671075567969074170075594330582, −7.81816281077375530964140174091, −5.90378252848219980159587084632, −5.63391517620560051384797623505, −5.06973831120185176441328984019, −3.97321214426304649442467496808, −1.57919323944505502932082854789,
0.951391273173875040061259424264, 2.65531844660784801280707580450, 4.19761712326911384018082606493, 5.17457673521491400357745480084, 6.37579071279781568049093814077, 6.97010583678693786037771369688, 7.65323152059431035708828148437, 9.825058830708649182552044935859, 10.57232857013528964803709220657, 11.09148437018318346683470475433
