| L(s) = 1 | + (−0.911 − 0.363i)2-s + (−1.72 − 0.187i)3-s + (−2.20 − 2.08i)4-s + (0.969 − 1.14i)5-s + (1.50 + 0.796i)6-s + (−4.53 − 2.73i)7-s + (2.89 + 6.26i)8-s + (2.92 + 0.644i)9-s + (−1.29 + 0.688i)10-s + (−5.02 − 0.272i)11-s + (3.40 + 4.00i)12-s + (2.47 + 11.2i)13-s + (3.14 + 4.13i)14-s + (−1.88 + 1.78i)15-s + (0.290 + 5.36i)16-s + (3.36 − 2.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.455 − 0.181i)2-s + (−0.573 − 0.0624i)3-s + (−0.551 − 0.522i)4-s + (0.193 − 0.228i)5-s + (0.250 + 0.132i)6-s + (−0.648 − 0.390i)7-s + (0.362 + 0.783i)8-s + (0.325 + 0.0716i)9-s + (−0.129 + 0.0688i)10-s + (−0.456 − 0.0247i)11-s + (0.283 + 0.334i)12-s + (0.190 + 0.866i)13-s + (0.224 + 0.295i)14-s + (−0.125 + 0.118i)15-s + (0.0181 + 0.335i)16-s + (0.197 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.179454 + 0.214752i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.179454 + 0.214752i\) |
| \(L(2)\) |
|
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 | \( 1 + (54.8 - 21.7i)T \) |
| good | 2 | \( 1 + (0.911 + 0.363i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (-0.969 + 1.14i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (4.53 + 2.73i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (5.02 + 0.272i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-2.47 - 11.2i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (-3.36 + 2.02i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (7.84 - 28.2i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (13.7 - 9.34i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (3.47 + 8.71i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (14.0 - 3.89i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (12.8 - 27.7i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (-20.7 + 30.5i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (7.43 - 0.402i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (-8.89 + 7.55i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (2.06 - 3.88i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (43.9 + 17.5i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-6.01 - 12.9i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (12.3 + 14.5i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (-39.0 - 51.4i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (101. - 11.0i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (35.3 + 104. i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (88.4 - 35.2i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (30.5 - 40.1i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.75113981013239710354822753103, −11.61705788174486125284284075777, −10.51523477136001594933456820416, −9.876404153634216043425953597705, −8.928623245477242760007271364827, −7.66502671975386593711485739084, −6.26416046886015273697310401177, −5.30567527769748665389129926183, −3.95232285913236607911693522308, −1.58112385591230519214692865941,
0.21598375099889367743276284320, 2.95326689695687702048757461242, 4.50927589006700913047931294134, 5.85397114846648586050903143074, 6.98650212275433748950000679710, 8.139524617805898687580354248878, 9.181445665273980597030957331556, 10.11226919749107273748469872409, 11.01469063650346623088132572508, 12.47500610466046653429197398388
