L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 − 2.59i)7-s + 0.999·8-s + (1 + 1.73i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s − 2·13-s + (−2 + 1.73i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (0.333 + 0.577i)9-s + (−0.158 + 0.273i)10-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s − 0.554·13-s + (−0.534 + 0.462i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0605332 - 0.953876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0605332 - 0.953876i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.884532617772672622808100804309, −9.144044935327297816535205476709, −8.195962417746597634886969137870, −7.37476008552137113706092814820, −6.85561272595111622785322484682, −5.14755307109228775436791498114, −4.30670245491343268400615565288, −3.13083680426458924304764141616, −1.90650933069563832207401384740, −0.51827449414454704169774619761,
1.93139348144554047455017181509, 3.43637722233213279630311974613, 4.32237046064832077499027145061, 5.66335509184139915189627053206, 6.29146568399933514754157951822, 7.35843489079672239310535681171, 8.229014046706414779608633266062, 9.024359342556082646431855598059, 9.796901175459375810086544189691, 10.34467220385752236620268643014