| L(s) = 1 | + (2.00 − 0.217i)2-s + (1.94 + 2.28i)3-s + (−3.85 + 0.848i)4-s + (−3.87 + 2.94i)5-s + (4.38 + 4.15i)6-s + (5.75 + 14.4i)7-s + (−22.7 + 7.67i)8-s + (−1.45 + 8.88i)9-s + (−7.10 + 6.73i)10-s + (−55.5 − 25.6i)11-s + (−9.42 − 7.16i)12-s + (−6.50 − 39.6i)13-s + (14.6 + 27.6i)14-s + (−14.2 − 3.13i)15-s + (−15.2 + 7.06i)16-s + (−12.6 + 31.6i)17-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.0769i)2-s + (0.373 + 0.440i)3-s + (−0.481 + 0.106i)4-s + (−0.346 + 0.263i)5-s + (0.298 + 0.282i)6-s + (0.310 + 0.779i)7-s + (−1.00 + 0.339i)8-s + (−0.0539 + 0.328i)9-s + (−0.224 + 0.212i)10-s + (−1.52 − 0.704i)11-s + (−0.226 − 0.172i)12-s + (−0.138 − 0.846i)13-s + (0.279 + 0.527i)14-s + (−0.245 − 0.0539i)15-s + (−0.238 + 0.110i)16-s + (−0.179 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.239070 + 1.03463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239070 + 1.03463i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.94 - 2.28i)T \) |
| 59 | \( 1 + (-37.1 - 451. i)T \) |
| good | 2 | \( 1 + (-2.00 + 0.217i)T + (7.81 - 1.71i)T^{2} \) |
| 5 | \( 1 + (3.87 - 2.94i)T + (33.4 - 120. i)T^{2} \) |
| 7 | \( 1 + (-5.75 - 14.4i)T + (-249. + 235. i)T^{2} \) |
| 11 | \( 1 + (55.5 + 25.6i)T + (861. + 1.01e3i)T^{2} \) |
| 13 | \( 1 + (6.50 + 39.6i)T + (-2.08e3 + 701. i)T^{2} \) |
| 17 | \( 1 + (12.6 - 31.6i)T + (-3.56e3 - 3.37e3i)T^{2} \) |
| 19 | \( 1 + (32.7 - 48.3i)T + (-2.53e3 - 6.37e3i)T^{2} \) |
| 23 | \( 1 + (-7.86 - 145. i)T + (-1.20e4 + 1.31e3i)T^{2} \) |
| 29 | \( 1 + (-57.7 - 6.27i)T + (2.38e4 + 5.24e3i)T^{2} \) |
| 31 | \( 1 + (20.3 + 30.0i)T + (-1.10e4 + 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-183. - 61.8i)T + (4.03e4 + 3.06e4i)T^{2} \) |
| 41 | \( 1 + (7.42 - 137. i)T + (-6.85e4 - 7.45e3i)T^{2} \) |
| 43 | \( 1 + (-43.8 + 20.2i)T + (5.14e4 - 6.05e4i)T^{2} \) |
| 47 | \( 1 + (154. + 117. i)T + (2.77e4 + 1.00e5i)T^{2} \) |
| 53 | \( 1 + (214. + 202. i)T + (8.06e3 + 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-48.6 + 5.29i)T + (2.21e5 - 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-277. + 93.6i)T + (2.39e5 - 1.82e5i)T^{2} \) |
| 71 | \( 1 + (111. + 84.9i)T + (9.57e4 + 3.44e5i)T^{2} \) |
| 73 | \( 1 + (-482. - 909. i)T + (-2.18e5 + 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-95.1 + 111. i)T + (-7.97e4 - 4.86e5i)T^{2} \) |
| 83 | \( 1 + (410. + 247. i)T + (2.67e5 + 5.05e5i)T^{2} \) |
| 89 | \( 1 + (-312. - 33.9i)T + (6.88e5 + 1.51e5i)T^{2} \) |
| 97 | \( 1 + (359. - 677. i)T + (-5.12e5 - 7.55e5i)T^{2} \) |
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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
−12.94195211324460909438787309804, −11.76194882345788502398287450945, −10.78058045874423467372598701995, −9.649838944729180270897460479108, −8.431968055198679657912500468192, −7.86853913940892679404229013353, −5.76606575357349645278975932401, −5.12200577117794816353576714534, −3.66596944147761938470927353016, −2.70037507900270806140154209678,
0.35412733820155459883004045117, 2.54831821668756117957489907674, 4.26133024612860127742465009351, 4.89479651689189994235603741972, 6.51863321547371280724256164542, 7.63294705358076846017687018137, 8.612321586349803241875877493631, 9.764599278967678360678987533866, 10.89114718741069545464569685045, 12.26211083168313885520639134596
