| L(s) = 1 | + (−2.50 − 0.349i)2-s + (−0.786 + 0.283i)3-s + (4.24 + 1.20i)4-s + (2.73 − 2.86i)5-s + (2.07 − 0.436i)6-s + (4.27 − 1.32i)7-s + (−5.59 − 2.46i)8-s + (−1.77 + 1.47i)9-s + (−7.87 + 6.23i)10-s + (−0.101 − 0.879i)11-s + (−3.68 + 0.255i)12-s + (1.65 + 4.27i)13-s + (−11.1 + 1.82i)14-s + (−1.33 + 3.03i)15-s + (5.65 + 3.50i)16-s + (0.609 − 4.07i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 − 0.247i)2-s + (−0.453 + 0.163i)3-s + (2.12 + 0.603i)4-s + (1.22 − 1.28i)5-s + (0.845 − 0.178i)6-s + (1.61 − 0.500i)7-s + (−1.97 − 0.873i)8-s + (−0.590 + 0.490i)9-s + (−2.48 + 1.97i)10-s + (−0.0307 − 0.265i)11-s + (−1.06 + 0.0737i)12-s + (0.459 + 1.18i)13-s + (−2.99 + 0.488i)14-s + (−0.345 + 0.782i)15-s + (1.41 + 0.875i)16-s + (0.147 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.629123 - 0.313466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629123 - 0.313466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.609 + 4.07i)T \) |
| good | 2 | \( 1 + (2.50 + 0.349i)T + (1.92 + 0.547i)T^{2} \) |
| 3 | \( 1 + (0.786 - 0.283i)T + (2.30 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-2.73 + 2.86i)T + (-0.230 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-4.27 + 1.32i)T + (5.77 - 3.95i)T^{2} \) |
| 11 | \( 1 + (0.101 + 0.879i)T + (-10.7 + 2.51i)T^{2} \) |
| 13 | \( 1 + (-1.65 - 4.27i)T + (-9.60 + 8.75i)T^{2} \) |
| 19 | \( 1 + (0.144 + 1.03i)T + (-18.2 + 5.19i)T^{2} \) |
| 23 | \( 1 + (-1.90 - 6.16i)T + (-18.9 + 12.9i)T^{2} \) |
| 29 | \( 1 + (0.474 + 0.376i)T + (6.63 + 28.2i)T^{2} \) |
| 31 | \( 1 + (1.50 + 0.0348i)T + (30.9 + 1.43i)T^{2} \) |
| 37 | \( 1 + (2.91 - 2.54i)T + (5.11 - 36.6i)T^{2} \) |
| 41 | \( 1 + (7.20 + 3.38i)T + (26.1 + 31.5i)T^{2} \) |
| 43 | \( 1 + (-5.24 - 1.23i)T + (38.4 + 19.1i)T^{2} \) |
| 47 | \( 1 + (-10.0 + 0.929i)T + (46.1 - 8.63i)T^{2} \) |
| 53 | \( 1 + (2.83 + 3.41i)T + (-9.73 + 52.0i)T^{2} \) |
| 59 | \( 1 + (4.35 + 2.98i)T + (21.3 + 55.0i)T^{2} \) |
| 61 | \( 1 + (-0.650 - 3.08i)T + (-55.8 + 24.6i)T^{2} \) |
| 67 | \( 1 + (-4.03 + 5.34i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-0.410 + 0.778i)T + (-40.1 - 58.5i)T^{2} \) |
| 73 | \( 1 + (10.1 - 7.27i)T + (23.1 - 69.2i)T^{2} \) |
| 79 | \( 1 + (-1.06 + 4.09i)T + (-69.0 - 38.4i)T^{2} \) |
| 83 | \( 1 + (5.08 - 0.235i)T + (82.6 - 7.65i)T^{2} \) |
| 89 | \( 1 + (4.45 - 11.5i)T + (-65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (10.6 + 5.59i)T + (54.8 + 80.0i)T^{2} \) |
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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
−11.36336588222839991547758652442, −10.67694905219883657716903253695, −9.594581237040382254564147415103, −8.887081265504004028893284619726, −8.251435366550946551489742490704, −7.15232920952257688529334242249, −5.65737605001248964977910151858, −4.77211586879432104421254827326, −2.05762138258245882496164858357, −1.15866242428529645085958054096,
1.53117740009496995202495139324, 2.67870712114540034048118750464, 5.58678655871653169460041068642, 6.17322410139887615858484057249, 7.23564288847965920750163468466, 8.274555480738717547230736661549, 8.974052150312661116096604089962, 10.27088824646938062408374525126, 10.71262311469285141484628902884, 11.30416295469342763859700278755
