| L(s) = 1 | + (−1.11 + 0.865i)2-s + (1.71 − 0.258i)3-s + (0.500 − 1.93i)4-s + (0.342 − 0.939i)5-s + (−1.69 + 1.77i)6-s + (−1.15 − 0.203i)7-s + (1.11 + 2.59i)8-s + (2.86 − 0.886i)9-s + (0.431 + 1.34i)10-s + (2.85 − 1.03i)11-s + (0.355 − 3.44i)12-s + (−4.28 + 3.59i)13-s + (1.46 − 0.773i)14-s + (0.342 − 1.69i)15-s + (−3.49 − 1.93i)16-s + (4.31 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.790 + 0.612i)2-s + (0.988 − 0.149i)3-s + (0.250 − 0.968i)4-s + (0.152 − 0.420i)5-s + (−0.690 + 0.723i)6-s + (−0.437 − 0.0770i)7-s + (0.395 + 0.918i)8-s + (0.955 − 0.295i)9-s + (0.136 + 0.425i)10-s + (0.859 − 0.312i)11-s + (0.102 − 0.994i)12-s + (−1.18 + 0.998i)13-s + (0.392 − 0.206i)14-s + (0.0884 − 0.438i)15-s + (−0.874 − 0.484i)16-s + (1.04 − 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44689 - 0.0255496i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44689 - 0.0255496i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 - 0.865i)T \) |
| 3 | \( 1 + (-1.71 + 0.258i)T \) |
| 5 | \( 1 + (-0.342 + 0.939i)T \) |
| good | 7 | \( 1 + (1.15 + 0.203i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.85 + 1.03i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.28 - 3.59i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.31 + 2.49i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.90 - 3.98i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.20 + 6.85i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 1.30i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.55 + 0.980i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (4.91 + 8.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.87 + 4.61i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.49 - 4.10i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.473 + 2.68i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.27iT - 53T^{2} \) |
| 59 | \( 1 + (-2.33 - 0.848i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.25 - 7.12i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.90 + 2.27i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.41 - 11.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.11 - 14.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.49 + 1.78i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.69 - 5.61i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.487 - 0.281i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.9 - 4.35i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.22652323037982276230166619182, −9.654061706523325690970115893144, −9.113103886397151464210621298507, −8.178791216067282068382438560485, −7.32636900892109797323698899838, −6.61508523596941594599942903427, −5.37751165892451864773691528339, −4.10965626756111994366303917515, −2.59371595631664231403237560671, −1.18016781125761405004784016955,
1.49644452053805748056277474124, 2.99861956393515224831572178484, 3.40582716146845756257451499413, 4.98919838386052760945828179922, 6.71012840383363834692151888986, 7.52546808557984816899173197757, 8.177927926433051511283460922666, 9.472432006502833901524961469141, 9.715620125182547549177949666376, 10.41943029522613378839730702507
