L(s) = 1 | + (0.140 + 0.0561i)2-s + (1.72 + 0.187i)3-s + (−2.88 − 2.73i)4-s + (1.82 − 2.14i)5-s + (0.232 + 0.123i)6-s + (−8.41 − 5.06i)7-s + (−0.507 − 1.09i)8-s + (2.92 + 0.644i)9-s + (0.377 − 0.200i)10-s + (−7.27 − 0.394i)11-s + (−4.45 − 5.25i)12-s + (−2.70 − 12.3i)13-s + (−0.901 − 1.18i)14-s + (3.54 − 3.35i)15-s + (0.851 + 15.7i)16-s + (−2.95 + 1.77i)17-s + ⋯ |
L(s) = 1 | + (0.0704 + 0.0280i)2-s + (0.573 + 0.0624i)3-s + (−0.721 − 0.683i)4-s + (0.364 − 0.429i)5-s + (0.0386 + 0.0205i)6-s + (−1.20 − 0.723i)7-s + (−0.0634 − 0.137i)8-s + (0.325 + 0.0716i)9-s + (0.0377 − 0.0200i)10-s + (−0.661 − 0.0358i)11-s + (−0.371 − 0.437i)12-s + (−0.208 − 0.946i)13-s + (−0.0643 − 0.0847i)14-s + (0.236 − 0.223i)15-s + (0.0532 + 0.981i)16-s + (−0.173 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.635919 - 0.986239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.635919 - 0.986239i\) |
\(L(2)\) |
|
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 | \( 1 + (57.4 + 13.2i)T \) |
good | 2 | \( 1 + (-0.140 - 0.0561i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (-1.82 + 2.14i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (8.41 + 5.06i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (7.27 + 0.394i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (2.70 + 12.3i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (2.95 - 1.77i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-2.43 + 8.78i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-32.9 + 22.3i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (18.1 + 45.5i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-33.0 + 9.16i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (22.3 - 48.2i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (5.08 - 7.50i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-39.5 + 2.14i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (-37.9 + 32.2i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (13.1 - 24.7i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (12.1 + 4.83i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-46.6 - 100. i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-52.6 - 61.9i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (13.3 + 17.5i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-65.5 + 7.13i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (5.55 + 16.4i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (-86.8 + 34.6i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (55.0 - 72.4i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.67969766878812485354142982577, −10.75624762114339746213567873026, −9.973920334223565738940600360846, −9.312914817674966790422118607904, −8.221640912296017047768657205086, −6.84688201486262090462002045270, −5.57929740857213957678021249893, −4.41592649463440819594179754265, −2.94397517788803774514454588580, −0.65282869124907571324225209210,
2.59297394517338907094019870958, 3.55518678393647597413154060140, 5.14412982378972751994367551683, 6.60433037886685270793324361309, 7.65846475973145150543150322125, 9.103624037947835580437545302011, 9.312177160362617026225789878722, 10.65614719146614224497035568307, 12.19840604760513889464124946073, 12.78535882671763450047418518825