L(s) = 1 | + (1.85 − 3.22i)2-s + (−2.70 + 1.29i)3-s + (−4.91 − 8.51i)4-s + (−4.36 + 2.43i)5-s + (−0.869 + 11.1i)6-s + (−4.87 − 5.02i)7-s − 21.7·8-s + (5.65 − 7.00i)9-s + (−0.272 + 18.5i)10-s + (10.0 − 5.82i)11-s + (24.3 + 16.7i)12-s − 9.22i·13-s + (−25.2 + 6.37i)14-s + (8.66 − 12.2i)15-s + (−20.7 + 35.8i)16-s + (1.56 + 2.70i)17-s + ⋯ |
L(s) = 1 | + (0.929 − 1.61i)2-s + (−0.902 + 0.431i)3-s + (−1.22 − 2.12i)4-s + (−0.873 + 0.487i)5-s + (−0.144 + 1.85i)6-s + (−0.696 − 0.717i)7-s − 2.71·8-s + (0.628 − 0.777i)9-s + (−0.0272 + 1.85i)10-s + (0.916 − 0.529i)11-s + (2.02 + 1.39i)12-s − 0.709i·13-s + (−1.80 + 0.455i)14-s + (0.577 − 0.816i)15-s + (−1.29 + 2.24i)16-s + (0.0919 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.148702 + 0.965312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.148702 + 0.965312i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.70 - 1.29i)T \) |
| 5 | \( 1 + (4.36 - 2.43i)T \) |
| 7 | \( 1 + (4.87 + 5.02i)T \) |
good | 2 | \( 1 + (-1.85 + 3.22i)T + (-2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-10.0 + 5.82i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 9.22iT - 169T^{2} \) |
| 17 | \( 1 + (-1.56 - 2.70i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-5.39 + 9.34i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (2.93 - 5.08i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 38.3iT - 841T^{2} \) |
| 31 | \( 1 + (15.7 + 27.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (20.0 + 11.5i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-30.0 + 52.1i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-27.9 - 48.4i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-23.2 + 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (19.8 - 34.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-86.4 + 49.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 62.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.5 + 21.6i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-15.5 + 26.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 93.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (34.2 + 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 119. iT - 9.40e3T^{2} \) |
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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
−12.57881647288034857843360884194, −11.85903492734455754762680212922, −10.92493469577396481952361169439, −10.43160234305585390386038920885, −9.245766564844852759354007588420, −6.89028809415216311446988549476, −5.51977209093637827140673443843, −4.04529476647648938905170817977, −3.39662783558458804280174924995, −0.62788813843130635130953740108,
3.95138980950879041366906497634, 5.05345128363409968311684225350, 6.24249054057629969230936734746, 7.02469518141656898301112494998, 8.138088184606921939551383007964, 9.384273760510351464414524736777, 11.70302602465683668901876685458, 12.26040698732709398950413299147, 13.02218777988649115660262994584, 14.21298579207547998031041588059