| L(s) = 1 | + (1.57 − 2.13i)2-s + (0.302 − 0.0572i)3-s + (−1.48 − 4.82i)4-s + (−2.22 − 0.238i)5-s + (0.354 − 0.735i)6-s + (2.53 + 0.757i)7-s + (−7.62 − 2.66i)8-s + (−2.70 + 1.06i)9-s + (−4.01 + 4.37i)10-s + (1.46 − 3.74i)11-s + (−0.725 − 1.37i)12-s + (3.98 + 0.449i)13-s + (5.61 − 4.22i)14-s + (−0.685 + 0.0551i)15-s + (−9.38 + 6.40i)16-s + (5.27 + 0.197i)17-s + ⋯ |
| L(s) = 1 | + (1.11 − 1.51i)2-s + (0.174 − 0.0330i)3-s + (−0.743 − 2.41i)4-s + (−0.994 − 0.106i)5-s + (0.144 − 0.300i)6-s + (0.958 + 0.286i)7-s + (−2.69 − 0.943i)8-s + (−0.901 + 0.353i)9-s + (−1.26 + 1.38i)10-s + (0.442 − 1.12i)11-s + (−0.209 − 0.396i)12-s + (1.10 + 0.124i)13-s + (1.50 − 1.12i)14-s + (−0.177 + 0.0142i)15-s + (−2.34 + 1.60i)16-s + (1.27 + 0.0478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585288 - 1.86616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585288 - 1.86616i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.22 + 0.238i)T \) |
| 7 | \( 1 + (-2.53 - 0.757i)T \) |
| good | 2 | \( 1 + (-1.57 + 2.13i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.302 + 0.0572i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.46 + 3.74i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 0.449i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-5.27 - 0.197i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (2.50 - 4.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0111 - 0.296i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-4.69 + 1.07i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.31 - 0.760i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (9.47 - 5.00i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-1.28 - 2.67i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.24 + 3.56i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (0.659 + 0.487i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (6.68 + 3.53i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.343 - 4.59i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.568 - 1.84i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-1.40 - 0.376i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.378 + 1.66i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (12.3 - 9.14i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-12.4 - 7.16i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.739 + 6.56i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (2.57 + 6.54i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (6.73 + 6.73i)T + 97iT^{2} \) |
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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
−11.72931566521906439241201770626, −11.13705020095665449790037582003, −10.39785006135930360085322637597, −8.737936215249298060078180397219, −8.199041978361541532529613890714, −6.04577396417512522803307618801, −5.15838267350694850819778474775, −3.88022231776838790962681015919, −3.12123043779866509195586317235, −1.36320213020852897539867865085,
3.32927742019531591510244074461, 4.28297969921792382838931906848, 5.22397897316147767128024100056, 6.49545875840701234357511001676, 7.41540196186467102485280096900, 8.197723635714463320859061318391, 8.928706111385768194334505173817, 10.93190878776599493136372337708, 11.93854162734737566870892151043, 12.53861739852731847479156727730
