| L(s) = 1 | + (1.84 − 1.29i)2-s + (1.67 + 0.444i)3-s + (1.05 − 2.88i)4-s + (−1.69 − 1.45i)5-s + (3.66 − 1.34i)6-s + (−1.72 − 0.461i)7-s + (−0.624 − 2.33i)8-s + (2.60 + 1.48i)9-s + (−5.01 − 0.500i)10-s + (4.43 + 2.56i)11-s + (3.04 − 4.36i)12-s + (−5.11 − 0.447i)13-s + (−3.77 + 1.37i)14-s + (−2.18 − 3.19i)15-s + (0.542 + 0.454i)16-s + (−5.53 + 3.87i)17-s + ⋯ |
| L(s) = 1 | + (1.30 − 0.913i)2-s + (0.966 + 0.256i)3-s + (0.525 − 1.44i)4-s + (−0.758 − 0.652i)5-s + (1.49 − 0.548i)6-s + (−0.651 − 0.174i)7-s + (−0.220 − 0.824i)8-s + (0.868 + 0.495i)9-s + (−1.58 − 0.158i)10-s + (1.33 + 0.772i)11-s + (0.877 − 1.26i)12-s + (−1.41 − 0.124i)13-s + (−1.00 + 0.367i)14-s + (−0.565 − 0.824i)15-s + (0.135 + 0.113i)16-s + (−1.34 + 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31652 - 1.60251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31652 - 1.60251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.67 - 0.444i)T \) |
| 5 | \( 1 + (1.69 + 1.45i)T \) |
| 19 | \( 1 + (-2.86 + 3.28i)T \) |
| good | 2 | \( 1 + (-1.84 + 1.29i)T + (0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (1.72 + 0.461i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-4.43 - 2.56i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.11 + 0.447i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (5.53 - 3.87i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (0.0277 - 0.0129i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.00676 - 0.0383i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.74 + 3.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.60 + 4.60i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.840 - 1.00i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.73 + 1.27i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (5.41 - 7.73i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (2.50 - 1.16i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-1.67 + 9.51i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.13 + 1.50i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.70 + 6.09i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (0.382 + 1.05i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.162 - 1.85i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-0.934 + 1.11i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-15.0 - 4.04i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-3.35 + 2.81i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (5.40 + 7.71i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−11.93053539892819138403339685064, −11.00102947503118683050038686791, −9.734995572692186302017544025292, −9.129437150677378637999496520117, −7.75126778329158009784264112427, −6.63173519376858102140867327793, −4.80035631600026225171855945898, −4.26324494720541867768398220851, −3.30271546902807055226657033085, −1.95941553674192227638356246978,
2.84883127279013443845080172210, 3.66256488950686308665814110733, 4.68303377214406369252288326780, 6.36653811342106070476547357433, 6.93043620958277788422620451898, 7.71070997040039433377693993095, 8.936435615438727409244595615844, 9.930851524547916515646809321704, 11.66950234845081054059029984694, 12.16408809146523736560350917764
