| L(s) = 1 | + (−2.06 − 0.820i)2-s + (1.72 + 0.187i)3-s + (0.667 + 0.632i)4-s + (4.71 − 5.55i)5-s + (−3.39 − 1.79i)6-s + (8.57 + 5.15i)7-s + (2.86 + 6.20i)8-s + (2.92 + 0.644i)9-s + (−14.2 + 7.57i)10-s + (−9.85 − 0.534i)11-s + (1.03 + 1.21i)12-s + (0.252 + 1.14i)13-s + (−13.4 − 17.6i)14-s + (9.16 − 8.67i)15-s + (−1.01 − 18.8i)16-s + (21.8 − 13.1i)17-s + ⋯ |
| L(s) = 1 | + (−1.03 − 0.410i)2-s + (0.573 + 0.0624i)3-s + (0.166 + 0.158i)4-s + (0.943 − 1.11i)5-s + (−0.565 − 0.299i)6-s + (1.22 + 0.736i)7-s + (0.358 + 0.775i)8-s + (0.325 + 0.0716i)9-s + (−1.42 + 0.757i)10-s + (−0.895 − 0.0485i)11-s + (0.0859 + 0.101i)12-s + (0.0194 + 0.0882i)13-s + (−0.959 − 1.26i)14-s + (0.610 − 0.578i)15-s + (−0.0637 − 1.17i)16-s + (1.28 − 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15178 - 0.612137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15178 - 0.612137i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.187i)T \) |
| 59 | \( 1 + (41.4 - 42.0i)T \) |
| good | 2 | \( 1 + (2.06 + 0.820i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (-4.71 + 5.55i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-8.57 - 5.15i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (9.85 + 0.534i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-0.252 - 1.14i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (-21.8 + 13.1i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (5.87 - 21.1i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-29.3 + 19.8i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (15.5 + 39.0i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (16.5 - 4.59i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (-6.01 + 12.9i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (6.55 - 9.66i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (28.9 - 1.57i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (63.5 - 53.9i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-15.7 + 29.7i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (-64.7 - 25.8i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (19.5 + 42.2i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (21.2 + 25.0i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (-63.0 - 82.8i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-89.2 + 9.70i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-29.4 - 87.3i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (110. - 44.1i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (41.1 - 54.1i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.23284681674871868960099834546, −11.08197132999861810877991237121, −9.962533486455019811310102093052, −9.306842877225667126161978876152, −8.370213010728726992505756012475, −7.88265455565627025040738806253, −5.52077656212364046748340889954, −4.90478216844296937542157711146, −2.37982958965538269246742414186, −1.30044604806099881712308487928,
1.59685327513147290445387225205, 3.33310607527659025087651143414, 5.16469224388712031170368662517, 6.87274250816620078931258873781, 7.54155856116680488374085313248, 8.430673835200756612073314121833, 9.587423162529899749316752669608, 10.46301188978962817117923806295, 11.02062603722368316519358923643, 12.97319762701088581677199013170
