| 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
−11.09148437018318346683470475433, −10.57232857013528964803709220657, −9.825058830708649182552044935859, −7.65323152059431035708828148437, −6.97010583678693786037771369688, −6.37579071279781568049093814077, −5.17457673521491400357745480084, −4.19761712326911384018082606493, −2.65531844660784801280707580450, −0.951391273173875040061259424264,
1.57919323944505502932082854789, 3.97321214426304649442467496808, 5.06973831120185176441328984019, 5.63391517620560051384797623505, 5.90378252848219980159587084632, 7.81816281077375530964140174091, 8.671075567969074170075594330582, 9.489324292954713316966780717672, 10.74566603444429881715419626286, 11.74954801609673732116103186911
