| L(s) = 1 | − 3-s − 5-s − 3.24i·7-s + 9-s − 5.82i·11-s − 3.24i·13-s + 15-s − 4.51·17-s + (−3.51 − 2.58i)19-s + 3.24i·21-s − 5.56i·23-s + 25-s − 27-s − 5.82i·29-s − 7.85·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.22i·7-s + 0.333·9-s − 1.75i·11-s − 0.899i·13-s + 0.258·15-s − 1.09·17-s + (−0.805 − 0.592i)19-s + 0.707i·21-s − 1.16i·23-s + 0.200·25-s − 0.192·27-s − 1.08i·29-s − 1.41·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7032121699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7032121699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (3.51 + 2.58i)T \) |
| good | 7 | \( 1 + 3.24iT - 7T^{2} \) |
| 11 | \( 1 + 5.82iT - 11T^{2} \) |
| 13 | \( 1 + 3.24iT - 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 23 | \( 1 + 5.56iT - 23T^{2} \) |
| 29 | \( 1 + 5.82iT - 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 - 0.255iT - 37T^{2} \) |
| 41 | \( 1 + 0.661iT - 41T^{2} \) |
| 43 | \( 1 - 0.255iT - 43T^{2} \) |
| 47 | \( 1 - 0.406iT - 47T^{2} \) |
| 53 | \( 1 - 5.56iT - 53T^{2} \) |
| 59 | \( 1 - 9.02T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 4.65T + 79T^{2} \) |
| 83 | \( 1 + 6.07iT - 83T^{2} \) |
| 89 | \( 1 + 3.64iT - 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−7.911356185606895757044027995430, −7.09942032447365261121531241538, −6.43502306575600544448391285828, −5.80660108889126323512003994408, −4.83796602447964915382789867982, −4.10401730123494150969906358395, −3.46586520692010833332600848957, −2.37772955929472110090827996593, −0.76694376556216454007553130977, −0.28993997801605106580078122689,
1.79373406968755348077345809360, 2.16824617437862267716294089307, 3.61048863443863980418941186053, 4.39765531388412966650138569710, 5.06913942371039168244362545539, 5.76750794032858877802422300500, 6.78262668326464707277240337471, 7.05653551323715648961789942331, 8.016895596572785766318306379421, 8.912767988397454009929194837763
