| L(s) = 1 | + (0.930 + 0.365i)2-s + (1.19 − 1.25i)3-s + (0.733 + 0.680i)4-s + (0.655 + 0.447i)5-s + (1.57 − 0.728i)6-s + (0.217 + 2.63i)7-s + (0.433 + 0.900i)8-s + (−0.136 − 2.99i)9-s + (0.447 + 0.655i)10-s + (−0.169 − 1.12i)11-s + (1.72 − 0.104i)12-s + (−0.0449 + 0.0358i)13-s + (−0.761 + 2.53i)14-s + (1.34 − 0.286i)15-s + (0.0747 + 0.997i)16-s + (−1.90 − 0.588i)17-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.258i)2-s + (0.690 − 0.723i)3-s + (0.366 + 0.340i)4-s + (0.293 + 0.199i)5-s + (0.641 − 0.297i)6-s + (0.0821 + 0.996i)7-s + (0.153 + 0.318i)8-s + (−0.0456 − 0.998i)9-s + (0.141 + 0.207i)10-s + (−0.0511 − 0.339i)11-s + (0.499 − 0.0301i)12-s + (−0.0124 + 0.00994i)13-s + (−0.203 + 0.677i)14-s + (0.347 − 0.0739i)15-s + (0.0186 + 0.249i)16-s + (−0.462 − 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29748 + 0.0616619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29748 + 0.0616619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.930 - 0.365i)T \) |
| 3 | \( 1 + (-1.19 + 1.25i)T \) |
| 7 | \( 1 + (-0.217 - 2.63i)T \) |
| good | 5 | \( 1 + (-0.655 - 0.447i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.169 + 1.12i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (0.0449 - 0.0358i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.90 + 0.588i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (2.58 - 1.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.62 + 5.25i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.688 + 0.157i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-5.31 - 3.06i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.45 - 2.28i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (8.84 - 4.25i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.62 - 3.18i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.56 - 6.52i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.289 - 0.312i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (5.09 - 3.47i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (7.26 + 7.83i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-4.84 + 8.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.9 - 3.17i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.46 + 3.32i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (1.41 + 2.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.28 - 10.3i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.00 - 0.603i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 3.54iT - 97T^{2} \) |
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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
−12.23266907540736803809650799976, −11.09530138397960839735147790505, −9.776265115750426632818076691956, −8.589243824121915103586483778343, −8.055676043655165398206788313738, −6.58524560576621184340777719555, −6.12078680836557035195483828391, −4.63909854114666015538503034975, −3.08654566779156881801610286301, −2.11342771377798297239559581443,
1.99505272481164238067100634008, 3.52867611362923660232503810068, 4.39090713200284047126644098283, 5.40788629675816348435825837920, 6.88629551843336077006732185746, 7.921171823962222705064248534677, 9.133263840637444034566919442808, 10.03891213363502275970531384841, 10.72456026632735856354996151125, 11.68115746684625786941788269235
