| L(s) = 1 | + (−2.19 − 1.36i)2-s + (−1.37 + 0.395i)3-s + (2.05 + 4.20i)4-s + (−2.03 − 0.933i)5-s + (3.56 + 1.02i)6-s + (1.36 − 2.36i)7-s + (0.724 − 6.89i)8-s + (−0.799 + 0.499i)9-s + (3.17 + 4.82i)10-s + (−2.40 + 2.66i)11-s + (−4.49 − 4.98i)12-s + (0.0645 + 1.84i)13-s + (−6.23 + 3.31i)14-s + (3.17 + 0.483i)15-s + (−5.26 + 6.73i)16-s + (0.321 − 0.0225i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.968i)2-s + (−0.795 + 0.228i)3-s + (1.02 + 2.10i)4-s + (−0.908 − 0.417i)5-s + (1.45 + 0.417i)6-s + (0.516 − 0.893i)7-s + (0.256 − 2.43i)8-s + (−0.266 + 0.166i)9-s + (1.00 + 1.52i)10-s + (−0.724 + 0.804i)11-s + (−1.29 − 1.44i)12-s + (0.0178 + 0.512i)13-s + (−1.66 + 0.885i)14-s + (0.818 + 0.124i)15-s + (−1.31 + 1.68i)16-s + (0.0780 − 0.00545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.290660 - 0.135467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.290660 - 0.135467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.03 + 0.933i)T \) |
| 19 | \( 1 + (4.02 - 1.67i)T \) |
| good | 2 | \( 1 + (2.19 + 1.36i)T + (0.876 + 1.79i)T^{2} \) |
| 3 | \( 1 + (1.37 - 0.395i)T + (2.54 - 1.58i)T^{2} \) |
| 7 | \( 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.40 - 2.66i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0645 - 1.84i)T + (-12.9 + 0.906i)T^{2} \) |
| 17 | \( 1 + (-0.321 + 0.0225i)T + (16.8 - 2.36i)T^{2} \) |
| 23 | \( 1 + (2.55 - 0.359i)T + (22.1 - 6.33i)T^{2} \) |
| 29 | \( 1 + (-3.37 - 0.236i)T + (28.7 + 4.03i)T^{2} \) |
| 31 | \( 1 + (-8.51 + 3.79i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.407 + 1.25i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.22 + 6.68i)T + (-9.91 - 39.7i)T^{2} \) |
| 43 | \( 1 + (-0.627 - 3.55i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.45 - 0.241i)T + (46.5 + 6.54i)T^{2} \) |
| 53 | \( 1 + (4.61 + 9.46i)T + (-32.6 + 41.7i)T^{2} \) |
| 59 | \( 1 + (0.613 + 1.51i)T + (-42.4 + 40.9i)T^{2} \) |
| 61 | \( 1 + (-0.426 + 0.0599i)T + (58.6 - 16.8i)T^{2} \) |
| 67 | \( 1 + (3.09 + 12.4i)T + (-59.1 + 31.4i)T^{2} \) |
| 71 | \( 1 + (-9.57 - 9.24i)T + (2.47 + 70.9i)T^{2} \) |
| 73 | \( 1 + (0.346 - 9.91i)T + (-72.8 - 5.09i)T^{2} \) |
| 79 | \( 1 + (-15.4 + 4.41i)T + (66.9 - 41.8i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 5.24i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-7.56 - 9.68i)T + (-21.5 + 86.3i)T^{2} \) |
| 97 | \( 1 + (0.505 - 2.02i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.70779897240321324071129446637, −10.33070766017681235040219256773, −9.245621292187969480476455032176, −8.023827983696801339568089481096, −7.88100673556581498075443933153, −6.62545565626436230884057217239, −4.82665416147442319501458047906, −3.94760533011111876025453523457, −2.27380377627392398275291957518, −0.67359905835992660239934528359,
0.65217430570256414949885130947, 2.71925072139540503306579964693, 4.95792406064989732045284625885, 5.99970078039444769239081814459, 6.52634793950187140493946804588, 7.75413583972271887324022739963, 8.311620522895032302800851226937, 8.945180832830304942741658203383, 10.37721992594859812068578356751, 10.86944845915264397553340507657
