| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−1.95 + 1.64i)3-s + (−0.939 + 0.342i)4-s + (0.968 + 0.352i)5-s + (1.95 + 1.64i)6-s + (−1.38 + 2.40i)7-s + (0.5 + 0.866i)8-s + (0.612 − 3.47i)9-s + (0.179 − 1.01i)10-s + (−0.5 − 0.866i)11-s + (1.27 − 2.21i)12-s + (−2.02 − 1.69i)13-s + (2.60 + 0.948i)14-s + (−2.47 + 0.900i)15-s + (0.766 − 0.642i)16-s + (−1.03 − 5.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−1.12 + 0.948i)3-s + (−0.469 + 0.171i)4-s + (0.433 + 0.157i)5-s + (0.798 + 0.670i)6-s + (−0.524 + 0.907i)7-s + (0.176 + 0.306i)8-s + (0.204 − 1.15i)9-s + (0.0566 − 0.321i)10-s + (−0.150 − 0.261i)11-s + (0.368 − 0.638i)12-s + (−0.560 − 0.470i)13-s + (0.696 + 0.253i)14-s + (−0.639 + 0.232i)15-s + (0.191 − 0.160i)16-s + (−0.250 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00112183 - 0.0769326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00112183 - 0.0769326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (2.12 - 3.80i)T \) |
| good | 3 | \( 1 + (1.95 - 1.64i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.968 - 0.352i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.38 - 2.40i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (2.02 + 1.69i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.03 + 5.86i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.40 - 0.874i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.371 - 2.10i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.106 - 0.184i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.84T + 37T^{2} \) |
| 41 | \( 1 + (7.49 - 6.28i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.16 + 1.15i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.10 + 11.9i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.79 + 1.37i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.678 - 3.84i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.75 + 2.82i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.420 + 2.38i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.82 - 1.02i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.751 + 0.630i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (7.19 - 6.03i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.29 - 9.17i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.02 - 7.57i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.28 + 7.26i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.86715815634646785640497626984, −10.58502714306894265758336893537, −10.05029236604293080182574249667, −9.409146951926984086730082251172, −8.313243535317499339299754780284, −6.70753461673954014310791807485, −5.58983710815480120888640398675, −5.07942715975006441731767070074, −3.69177252883000555629656081811, −2.39145163237555738161274620699,
0.05669358788628023989346278406, 1.75423139408868518827758733148, 4.06632618839332731580096383919, 5.26646134709416658626559811348, 6.23895306638505556951692361301, 6.84099237853237145571731309632, 7.57496747289220609941542281278, 8.795210409904515072518809936164, 9.979239593598435232321810340343, 10.66210871686895035196740996587
