| L(s) = 1 | + (−1.64 − 0.539i)3-s + (−1.30 − 1.81i)5-s + (−2.19 + 1.47i)7-s + (2.41 + 1.77i)9-s + 0.958i·11-s − 0.157·13-s + (1.16 + 3.69i)15-s − 2.51i·17-s + 1.98i·19-s + (4.41 − 1.23i)21-s − 2.67·23-s + (−1.60 + 4.73i)25-s + (−3.02 − 4.22i)27-s − 1.25i·29-s + 8.66i·31-s + ⋯ |
| L(s) = 1 | + (−0.950 − 0.311i)3-s + (−0.582 − 0.812i)5-s + (−0.831 + 0.555i)7-s + (0.806 + 0.591i)9-s + 0.288i·11-s − 0.0436·13-s + (0.300 + 0.953i)15-s − 0.609i·17-s + 0.454i·19-s + (0.963 − 0.269i)21-s − 0.557·23-s + (−0.321 + 0.946i)25-s + (−0.582 − 0.813i)27-s − 0.232i·29-s + 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7612196025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7612196025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.64 + 0.539i)T \) |
| 5 | \( 1 + (1.30 + 1.81i)T \) |
| 7 | \( 1 + (2.19 - 1.47i)T \) |
| good | 11 | \( 1 - 0.958iT - 11T^{2} \) |
| 13 | \( 1 + 0.157T + 13T^{2} \) |
| 17 | \( 1 + 2.51iT - 17T^{2} \) |
| 19 | \( 1 - 1.98iT - 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 1.25iT - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 2.29iT - 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 6.58iT - 43T^{2} \) |
| 47 | \( 1 - 5.60iT - 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.27iT - 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 9.75iT - 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 + 0.517T + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.954T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.301881711514189476873727616342, −8.472248089637252883632349165297, −7.57245980482963631611176585920, −6.84308614445169789448089018423, −5.96196747998859191992909075954, −5.23116011753849623886462785998, −4.45176105496932216315266114063, −3.38532209482748731895875795693, −1.94739353027625459270721104326, −0.56591660448229970397141938347,
0.68160766648316481616988575757, 2.57551538324143172405170428147, 3.82979155292808169066393307254, 4.14837837030374023700919820679, 5.54661133225979102115026539766, 6.25227168911992597764378105695, 6.94812770113853110570520213907, 7.59347207800766028801092971756, 8.669417750390517618433049515279, 9.876249508101202644075087573550
