| L(s) = 1 | + (−1.42 + 1.03i)2-s + (−0.104 + 0.994i)3-s + (0.336 − 1.03i)4-s + (0.5 − 0.866i)5-s + (−0.878 − 1.52i)6-s + (1.53 − 0.326i)7-s + (−0.495 − 1.52i)8-s + (−0.978 − 0.207i)9-s + (0.183 + 1.74i)10-s + (1.12 + 1.24i)11-s + (0.993 + 0.442i)12-s + (2.31 − 1.03i)13-s + (−1.84 + 2.05i)14-s + (0.809 + 0.587i)15-s + (4.03 + 2.93i)16-s + (2.66 − 2.96i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 + 0.730i)2-s + (−0.0603 + 0.574i)3-s + (0.168 − 0.517i)4-s + (0.223 − 0.387i)5-s + (−0.358 − 0.621i)6-s + (0.580 − 0.123i)7-s + (−0.175 − 0.538i)8-s + (−0.326 − 0.0693i)9-s + (0.0580 + 0.552i)10-s + (0.338 + 0.376i)11-s + (0.286 + 0.127i)12-s + (0.641 − 0.285i)13-s + (−0.493 + 0.548i)14-s + (0.208 + 0.151i)15-s + (1.00 + 0.733i)16-s + (0.646 − 0.718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.746861 + 0.547173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746861 + 0.547173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.10 - 2.22i)T \) |
| good | 2 | \( 1 + (1.42 - 1.03i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (-1.53 + 0.326i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 1.24i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.31 + 1.03i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.66 + 2.96i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.43 - 1.97i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.958 + 2.95i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.61 - 2.62i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-1.93 - 3.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.554 - 5.27i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (1.39 + 0.620i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-4.68 - 3.40i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (11.0 + 2.35i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.124 + 1.18i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 + (-2.38 + 4.13i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.95 - 1.47i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-3.72 - 4.13i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (3.88 - 4.31i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.475 - 4.52i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.77 + 14.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.893 + 2.75i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.97755111232768445605079491958, −9.873080531881093667402253543331, −9.496881545163616784525362413007, −8.419085519956273416145493691539, −7.86724384028531990959502018326, −6.76607766040698471874388639154, −5.69562267680707852161153879569, −4.63597872668224461555883635098, −3.33971706979124233299503758853, −1.16235069740214089962174938566,
1.10872655637050880399180457862, 2.18987871564712764330556516419, 3.54335592910441531446994816132, 5.35450775213082866535552584547, 6.22468757343320431448730484219, 7.54420555730914791531912534893, 8.252093923515071698228607019411, 9.166253279931736233989964857765, 9.948152860334509823422805185959, 10.98347821502062224323867503248
