L(s) = 1 | + (1.04 + 0.949i)2-s + (−0.0504 + 0.188i)3-s + (0.198 + 1.99i)4-s + (−2.13 − 0.657i)5-s + (−0.231 + 0.149i)6-s + (−1.68 + 2.27i)8-s + (2.56 + 1.48i)9-s + (−1.61 − 2.71i)10-s + (2.70 − 1.56i)11-s + (−0.384 − 0.0631i)12-s + (−2.50 + 2.50i)13-s + (0.231 − 0.369i)15-s + (−3.92 + 0.788i)16-s + (−1.04 + 3.90i)17-s + (1.28 + 3.98i)18-s + (−1.64 + 2.85i)19-s + ⋯ |
L(s) = 1 | + (0.741 + 0.671i)2-s + (−0.0291 + 0.108i)3-s + (0.0990 + 0.995i)4-s + (−0.955 − 0.294i)5-s + (−0.0945 + 0.0610i)6-s + (−0.594 + 0.804i)8-s + (0.855 + 0.493i)9-s + (−0.511 − 0.859i)10-s + (0.816 − 0.471i)11-s + (−0.111 − 0.0182i)12-s + (−0.694 + 0.694i)13-s + (0.0598 − 0.0953i)15-s + (−0.980 + 0.197i)16-s + (−0.253 + 0.947i)17-s + (0.302 + 0.939i)18-s + (−0.377 + 0.653i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.348051 + 1.55859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.348051 + 1.55859i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.04 - 0.949i)T \) |
| 5 | \( 1 + (2.13 + 0.657i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.0504 - 0.188i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.70 + 1.56i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.50 - 2.50i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.04 - 3.90i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.64 - 2.85i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.79 - 2.35i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.55iT - 29T^{2} \) |
| 31 | \( 1 + (-7.50 + 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.97 - 1.33i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + (1.17 + 1.17i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.218 - 0.817i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.53 - 0.411i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.20 + 9.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.40 - 5.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.33 - 2.50i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 3.15iT - 71T^{2} \) |
| 73 | \( 1 + (-8.20 - 2.19i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.0767 + 0.132i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.57 - 8.57i)T + 83iT^{2} \) |
| 89 | \( 1 + (-15.0 - 8.70i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.28 + 2.28i)T + 97iT^{2} \) |
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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
−10.42147049680368813517821095249, −9.369450487049111935083943991302, −8.305676447236949012653388415198, −7.88133428552852358233546559849, −6.88921629496101289330604426504, −6.17852209478583978748489468777, −4.96961535735792072016549953930, −4.13419576759975459831516732860, −3.67568944767264617726404523460, −1.93160697578922178559046777987,
0.57301518333029801746347796701, 2.18952442731059498584882565516, 3.34364586538992645235469703892, 4.28336583993403096532410884540, 4.83873247954729607278850677314, 6.33123589827008374549671831086, 6.89782551582704696574732173982, 7.80969560456039738829758145291, 9.022695208415656978013214239022, 9.951844380947987316462947930186