L(s) = 1 | + (−0.866 + 1.5i)3-s + (2.09 − 3.63i)5-s + 4.28·7-s + 4.71·11-s + (−0.615 − 1.06i)13-s + (3.63 + 6.29i)15-s + (−1.61 + 2.79i)17-s + (−0.625 − 4.31i)19-s + (−3.71 + 6.43i)21-s + (1.90 + 3.29i)23-s + (−6.31 − 10.9i)25-s − 5.19·27-s + (−2.09 − 3.63i)29-s + 0.825·31-s + (−4.08 + 7.07i)33-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (0.938 − 1.62i)5-s + 1.62·7-s + 1.42·11-s + (−0.170 − 0.295i)13-s + (0.938 + 1.62i)15-s + (−0.391 + 0.678i)17-s + (−0.143 − 0.989i)19-s + (−0.810 + 1.40i)21-s + (0.397 + 0.687i)23-s + (−1.26 − 2.18i)25-s − 1.00·27-s + (−0.389 − 0.675i)29-s + 0.148·31-s + (−0.710 + 1.23i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.127743415\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.127743415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.625 + 4.31i)T \) |
good | 3 | \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.09 + 3.63i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 + (0.615 + 1.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.61 - 2.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.90 - 3.29i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.09 + 3.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.825T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + (-2.69 + 4.67i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.49 - 2.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.653 + 1.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.38 + 2.39i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.31 - 12.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.09 - 7.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.69 - 2.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.04 - 7.01i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.69 - 8.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.31 + 4.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 + (5.61 + 9.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.92 - 10.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.635625732743177770997310910879, −8.900637335652653662485985108021, −8.467808294542287608557160017106, −7.32101097930249377759373064582, −5.91540048374840047871651602628, −5.35064817680255238524958022967, −4.53417916606994304312776649660, −4.20446982534396736646300732814, −1.98088980597585367668377714937, −1.18128176413255367918321264832,
1.48033238974674567267175112715, 2.06169113165797556832147223520, 3.47064726783815038385657704151, 4.72813100263886450900511061483, 5.86452808009993485213285628680, 6.54417106974782793043651361216, 7.05397920702501755410134905023, 7.83551857615272581042419834344, 9.013457399482249135790495626722, 9.807481959652829483841254168917