| L(s) = 1 | + (2.36 − 3.49i)2-s + (3.40 + 3.92i)3-s + (−3.62 − 9.10i)4-s + (−8.13 + 17.5i)5-s + (21.7 − 2.57i)6-s + (0.277 + 0.998i)7-s + (−7.43 − 1.63i)8-s + (−3.86 + 26.7i)9-s + (42.1 + 70.0i)10-s + (−20.7 − 19.6i)11-s + (23.4 − 45.2i)12-s + (−5.93 + 54.5i)13-s + (4.14 + 1.39i)14-s + (−96.8 + 27.8i)15-s + (33.6 − 31.8i)16-s + (50.0 + 13.8i)17-s + ⋯ |
| L(s) = 1 | + (0.837 − 1.23i)2-s + (0.654 + 0.756i)3-s + (−0.453 − 1.13i)4-s + (−0.728 + 1.57i)5-s + (1.48 − 0.175i)6-s + (0.0149 + 0.0538i)7-s + (−0.328 − 0.0723i)8-s + (−0.143 + 0.989i)9-s + (1.33 + 2.21i)10-s + (−0.568 − 0.538i)11-s + (0.563 − 1.08i)12-s + (−0.126 + 1.16i)13-s + (0.0790 + 0.0266i)14-s + (−1.66 + 0.479i)15-s + (0.525 − 0.497i)16-s + (0.714 + 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.67728 + 0.679203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.67728 + 0.679203i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.40 - 3.92i)T \) |
| 59 | \( 1 + (-396. + 220. i)T \) |
| good | 2 | \( 1 + (-2.36 + 3.49i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (8.13 - 17.5i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-0.277 - 0.998i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (20.7 + 19.6i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (5.93 - 54.5i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-50.0 - 13.8i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-122. - 93.0i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (31.8 + 60.0i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-58.5 + 39.6i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-12.7 - 16.7i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (19.6 + 89.4i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (448. + 237. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-29.5 - 31.1i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (148. - 68.7i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-310. + 515. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-687. - 465. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (119. - 545. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-228. - 492. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-31.2 + 92.7i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-18.7 + 346. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (2.89 - 17.6i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (342. + 504. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (82.0 + 243. i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.89435173989921329231014842721, −11.47360971182248226971937764827, −10.34128941136646465530664301324, −10.02508857970285959371063474625, −8.267172801418303640696027910480, −7.19172204498031886147528745273, −5.41543613567713011932773194311, −3.91366517968504163047466882988, −3.35464483134061964385715467884, −2.29039827608148654178691523132,
0.954758805307666860548007862044, 3.36832755994218507617643766070, 4.82550211025288808638193412010, 5.52853262671260244989850611523, 7.17892909546755438971889255702, 7.80866760777020409514614134008, 8.507058047795669672012498328878, 9.776856893138799810144635199162, 11.83839141664449918521594823409, 12.57448795919204477356769441065
