| L(s) = 1 | + (−0.120 − 1.40i)2-s + (0.857 + 0.514i)3-s + (−1.97 + 0.340i)4-s + (−0.180 + 0.0645i)5-s + (0.620 − 1.27i)6-s + (−2.44 + 0.742i)7-s + (0.717 + 2.73i)8-s + (0.471 + 0.881i)9-s + (0.112 + 0.246i)10-s + (0.138 − 0.186i)11-s + (−1.86 − 0.721i)12-s + (0.567 − 0.268i)13-s + (1.34 + 3.35i)14-s + (−0.188 − 0.0373i)15-s + (3.76 − 1.34i)16-s + (−7.12 + 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.0853 − 0.996i)2-s + (0.495 + 0.296i)3-s + (−0.985 + 0.170i)4-s + (−0.0807 + 0.0288i)5-s + (0.253 − 0.518i)6-s + (−0.924 + 0.280i)7-s + (0.253 + 0.967i)8-s + (0.157 + 0.293i)9-s + (0.0356 + 0.0779i)10-s + (0.0417 − 0.0562i)11-s + (−0.538 − 0.208i)12-s + (0.157 − 0.0744i)13-s + (0.358 + 0.897i)14-s + (−0.0485 − 0.00965i)15-s + (0.942 − 0.335i)16-s + (−1.72 + 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.610996 + 0.440000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.610996 + 0.440000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.120 + 1.40i)T \) |
| 3 | \( 1 + (-0.857 - 0.514i)T \) |
| good | 5 | \( 1 + (0.180 - 0.0645i)T + (3.86 - 3.17i)T^{2} \) |
| 7 | \( 1 + (2.44 - 0.742i)T + (5.82 - 3.88i)T^{2} \) |
| 11 | \( 1 + (-0.138 + 0.186i)T + (-3.19 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-0.567 + 0.268i)T + (8.24 - 10.0i)T^{2} \) |
| 17 | \( 1 + (7.12 - 1.41i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-0.0680 - 1.38i)T + (-18.9 + 1.86i)T^{2} \) |
| 23 | \( 1 + (-0.435 - 4.42i)T + (-22.5 + 4.48i)T^{2} \) |
| 29 | \( 1 + (1.14 - 7.74i)T + (-27.7 - 8.41i)T^{2} \) |
| 31 | \( 1 + (-2.74 - 1.13i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-6.30 - 6.96i)T + (-3.62 + 36.8i)T^{2} \) |
| 41 | \( 1 + (7.22 + 5.92i)T + (7.99 + 40.2i)T^{2} \) |
| 43 | \( 1 + (1.82 + 3.04i)T + (-20.2 + 37.9i)T^{2} \) |
| 47 | \( 1 + (-7.20 - 4.81i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (0.758 + 5.11i)T + (-50.7 + 15.3i)T^{2} \) |
| 59 | \( 1 + (3.89 + 1.84i)T + (37.4 + 45.6i)T^{2} \) |
| 61 | \( 1 + (6.52 - 1.63i)T + (53.7 - 28.7i)T^{2} \) |
| 67 | \( 1 + (-1.61 - 6.44i)T + (-59.0 + 31.5i)T^{2} \) |
| 71 | \( 1 + (3.56 + 1.90i)T + (39.4 + 59.0i)T^{2} \) |
| 73 | \( 1 + (-4.81 - 1.46i)T + (60.6 + 40.5i)T^{2} \) |
| 79 | \( 1 + (-2.21 - 3.31i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-3.70 + 4.09i)T + (-8.13 - 82.6i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 15.9i)T + (-87.2 - 17.3i)T^{2} \) |
| 97 | \( 1 + (-2.43 + 5.86i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−10.46470600939765576322521164170, −9.622985756393691630255393817863, −9.021585892263661234846784723413, −8.323750105145279973134469965011, −7.14402568591991184890203498358, −5.95598987911867888533477708298, −4.78498164334222811249084058337, −3.74011989539932876313126600542, −3.00077018923105252816222005514, −1.77914226354429591673477467398,
0.35988960011212655346141111496, 2.47569546600917666002257897875, 3.90349977794956470021423976196, 4.64946952330192715683667775542, 6.20617530472212475825196267399, 6.55651308851178142786225932597, 7.54011764658351225978489244269, 8.346107758732900651904672133011, 9.204525715815894768306652094435, 9.759925455247914996909361679090
