| L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.167 + 1.72i)3-s + (0.173 + 0.984i)4-s + (2.66 + 0.971i)5-s + (−1.23 + 1.21i)6-s + (−0.173 + 0.984i)7-s + (−0.500 + 0.866i)8-s + (−2.94 − 0.576i)9-s + (1.41 + 2.45i)10-s + (1.35 − 0.492i)11-s + (−1.72 + 0.134i)12-s + (2.58 − 2.17i)13-s + (−0.766 + 0.642i)14-s + (−2.12 + 4.43i)15-s + (−0.939 + 0.342i)16-s + (−3.09 − 5.36i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.0966 + 0.995i)3-s + (0.0868 + 0.492i)4-s + (1.19 + 0.434i)5-s + (−0.504 + 0.495i)6-s + (−0.0656 + 0.372i)7-s + (−0.176 + 0.306i)8-s + (−0.981 − 0.192i)9-s + (0.448 + 0.777i)10-s + (0.407 − 0.148i)11-s + (−0.498 + 0.0388i)12-s + (0.717 − 0.602i)13-s + (−0.204 + 0.171i)14-s + (−0.547 + 1.14i)15-s + (−0.234 + 0.0855i)16-s + (−0.751 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15491 + 1.65049i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15491 + 1.65049i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.167 - 1.72i)T \) |
| 7 | \( 1 + (0.173 - 0.984i)T \) |
| good | 5 | \( 1 + (-2.66 - 0.971i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.35 + 0.492i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.58 + 2.17i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.09 + 5.36i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.565 - 3.20i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.891 + 0.748i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.986 + 5.59i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.720 - 1.24i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.30 - 1.93i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.17 + 2.97i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.21 - 12.5i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 + (6.06 + 2.20i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 8.68i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.41 + 7.90i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.170 - 0.294i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.05 + 8.74i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.4 - 8.75i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.05 + 0.886i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 7.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.40 - 1.60i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.44276565388985821185725302506, −10.80941042111267073029777393676, −9.636339673501705357522641760985, −9.171460794144906690420795614557, −7.934764030095052249759218611293, −6.41395803952520125143242990001, −5.87420018219264400346738360694, −4.93488307511556417497811979386, −3.64029459256264657356804800504, −2.48503873046617358567869999577,
1.36613283990483333280779761658, 2.30360484312342141509262808807, 4.00562911108333744533332812508, 5.33559027637192634873998355207, 6.31597590412807372721839875578, 6.86419638198961499783785514648, 8.512268565814877054228230222399, 9.160459083380422971986985952542, 10.44253063646023349809734456636, 11.13211946208888471630166931966
