| L(s) = 1 | + (0.865 + 0.606i)2-s + (−0.905 − 1.47i)3-s + (−0.301 − 0.829i)4-s + (2.13 + 0.660i)5-s + (0.111 − 1.82i)6-s + (−1.22 − 2.61i)7-s + (0.788 − 2.94i)8-s + (−1.36 + 2.67i)9-s + (1.44 + 1.86i)10-s + (1.51 + 1.80i)11-s + (−0.951 + 1.19i)12-s + (3.11 + 4.45i)13-s + (0.530 − 3.00i)14-s + (−0.958 − 3.75i)15-s + (1.11 − 0.935i)16-s + (−0.716 − 2.67i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.428i)2-s + (−0.522 − 0.852i)3-s + (−0.150 − 0.414i)4-s + (0.955 + 0.295i)5-s + (0.0455 − 0.746i)6-s + (−0.461 − 0.990i)7-s + (0.278 − 1.04i)8-s + (−0.453 + 0.891i)9-s + (0.458 + 0.590i)10-s + (0.457 + 0.545i)11-s + (−0.274 + 0.345i)12-s + (0.865 + 1.23i)13-s + (0.141 − 0.804i)14-s + (−0.247 − 0.968i)15-s + (0.278 − 0.233i)16-s + (−0.173 − 0.648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25088 - 0.425893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25088 - 0.425893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.905 + 1.47i)T \) |
| 5 | \( 1 + (-2.13 - 0.660i)T \) |
| good | 2 | \( 1 + (-0.865 - 0.606i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.22 + 2.61i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 1.80i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.11 - 4.45i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.716 + 2.67i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.35 - 3.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.646 - 0.301i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 3.53i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.18 - 1.88i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.26 - 1.14i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.70 - 1.00i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.126 + 1.44i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-9.40 + 4.38i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (7.22 + 7.22i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.27 - 5.26i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 0.960i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.11 + 4.28i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (4.24 + 2.44i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.73 + 1.80i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (12.8 - 2.26i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.73 - 3.90i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.33 - 5.77i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (18.2 + 1.59i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.35735006884153109378790911165, −12.51814496309778200418647564275, −11.02287245402125740198828122418, −10.20330670041426121948098173678, −9.025351451370634879033353567279, −6.94513819767713345055791334655, −6.69806782127300872008800452450, −5.57274395656259596082607425569, −4.14225943850411565212918298631, −1.61084852062237524839067963488,
2.76146994797235412411978617918, 4.10985358826279372378694105438, 5.53213477091754911372154321416, 6.12049925247612875375243373557, 8.555727978663413800076645158798, 9.114426993004912496151265636563, 10.51452502583358118319523759647, 11.28763856708171091441683446721, 12.59284588875209539566788779859, 12.96967677607565314336035371608
