| L(s) = 1 | + (−1.11 + 0.871i)2-s + (0.593 − 1.62i)3-s + (0.482 − 1.94i)4-s + (3.29 − 1.75i)5-s + (0.757 + 2.32i)6-s + (−0.370 + 0.0737i)7-s + (1.15 + 2.58i)8-s + (−2.29 − 1.93i)9-s + (−2.13 + 4.82i)10-s + (4.94 + 4.05i)11-s + (−2.87 − 1.93i)12-s + (2.85 + 1.52i)13-s + (0.348 − 0.405i)14-s + (−0.911 − 6.40i)15-s + (−3.53 − 1.87i)16-s + (−1.84 − 0.762i)17-s + ⋯ |
| L(s) = 1 | + (−0.787 + 0.616i)2-s + (0.342 − 0.939i)3-s + (0.241 − 0.970i)4-s + (1.47 − 0.787i)5-s + (0.309 + 0.951i)6-s + (−0.140 + 0.0278i)7-s + (0.408 + 0.912i)8-s + (−0.765 − 0.643i)9-s + (−0.675 + 1.52i)10-s + (1.49 + 1.22i)11-s + (−0.829 − 0.558i)12-s + (0.791 + 0.422i)13-s + (0.0932 − 0.108i)14-s + (−0.235 − 1.65i)15-s + (−0.883 − 0.467i)16-s + (−0.446 − 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27947 - 0.439302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27947 - 0.439302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 - 0.871i)T \) |
| 3 | \( 1 + (-0.593 + 1.62i)T \) |
| good | 5 | \( 1 + (-3.29 + 1.75i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.370 - 0.0737i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-4.94 - 4.05i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.85 - 1.52i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.84 + 0.762i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.60 + 5.28i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (6.74 - 4.50i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.52 + 1.24i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (0.401 + 0.401i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.65 + 5.44i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (0.603 - 0.403i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.837 - 8.50i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (0.573 + 0.237i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.48 - 2.85i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (11.0 - 5.91i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.569 + 5.78i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (8.13 + 0.801i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.18 + 0.633i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.359 + 1.80i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.60 + 3.87i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.559 - 1.84i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (2.82 - 4.23i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.32 + 1.32i)T - 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.22312264442022455121913079047, −9.730543892680315466720929653427, −9.290478997259134078802304070562, −8.703236379638069395504136046804, −7.43667359814178522430155472486, −6.41849245526783434857835096802, −6.04280181657603685380802503858, −4.56305285949648076088308157035, −2.10747649733692410691674580158, −1.36874145803439441177738592409,
1.81966244602754516858379870973, 3.13393675495097061516584166691, 3.97354291564381546146302056544, 5.95260734709073011539150885648, 6.47937088660757718184602372923, 8.281164975188409417109833978092, 8.859426057169604768428232094378, 9.787358327638569282503080885289, 10.39406897714229093655240967572, 10.98084584127648167502995010652
