L(s) = 1 | + (−1.40 − 0.183i)2-s + (−0.908 − 0.524i)3-s + (1.93 + 0.514i)4-s + (0.5 + 0.866i)5-s + (1.17 + 0.901i)6-s + (−2.14 + 1.54i)7-s + (−2.61 − 1.07i)8-s + (−0.949 − 1.64i)9-s + (−0.542 − 1.30i)10-s + (−1.17 + 2.03i)11-s + (−1.48 − 1.48i)12-s + 1.21·13-s + (3.29 − 1.77i)14-s − 1.04i·15-s + (3.47 + 1.98i)16-s + (−4.23 − 2.44i)17-s + ⋯ |
L(s) = 1 | + (−0.991 − 0.129i)2-s + (−0.524 − 0.302i)3-s + (0.966 + 0.257i)4-s + (0.223 + 0.387i)5-s + (0.480 + 0.368i)6-s + (−0.811 + 0.583i)7-s + (−0.924 − 0.380i)8-s + (−0.316 − 0.548i)9-s + (−0.171 − 0.413i)10-s + (−0.354 + 0.614i)11-s + (−0.428 − 0.427i)12-s + 0.336·13-s + (0.880 − 0.473i)14-s − 0.270i·15-s + (0.867 + 0.497i)16-s + (−1.02 − 0.593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0633256 + 0.176985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0633256 + 0.176985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.183i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.14 - 1.54i)T \) |
good | 3 | \( 1 + (0.908 + 0.524i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.17 - 2.03i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + (4.23 + 2.44i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.21 - 1.27i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.59 - 4.38i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.21iT - 29T^{2} \) |
| 31 | \( 1 + (1.68 - 2.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.16 - 3.55i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.18iT - 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + (5.01 + 8.67i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.03 - 4.05i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.90 + 5.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.560 + 0.971i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.386 + 0.670i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.53iT - 71T^{2} \) |
| 73 | \( 1 + (-11.1 - 6.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.34 + 0.776i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.47iT - 83T^{2} \) |
| 89 | \( 1 + (9.58 - 5.53i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−12.12988784648330216941649970859, −11.20389222273909238619696276495, −10.28519796410211664939472660241, −9.398938581538731986872251570207, −8.616245766500614296233031589718, −7.22451800971517407571899032264, −6.49381106133427862088895148293, −5.63773267672784554688636960148, −3.44726223608499805852609932288, −2.08230526102978484763373529619,
0.18881193094392034756360007709, 2.37273327250889114553447485825, 4.20308456559035650421699972474, 5.83606001397488189424325654060, 6.39428963192634129910889790589, 7.81678959290296928206710036294, 8.637301657121314060800308064630, 9.667439289453808609142177384262, 10.62579813115061503251463154483, 10.99830360687118193825230630952