| L(s) = 1 | + (1.33 − 0.454i)2-s + (0.322 − 1.70i)3-s + (1.58 − 1.21i)4-s + (−0.205 + 0.307i)5-s + (−0.342 − 2.42i)6-s + (1.38 + 3.34i)7-s + (1.56 − 2.35i)8-s + (−2.79 − 1.09i)9-s + (−0.135 + 0.505i)10-s + (−4.03 + 0.802i)11-s + (−1.56 − 3.09i)12-s + (−1.78 + 1.19i)13-s + (3.37 + 3.85i)14-s + (0.457 + 0.448i)15-s + (1.03 − 3.86i)16-s + (4.80 − 4.80i)17-s + ⋯ |
| L(s) = 1 | + (0.946 − 0.321i)2-s + (0.186 − 0.982i)3-s + (0.793 − 0.609i)4-s + (−0.0918 + 0.137i)5-s + (−0.139 − 0.990i)6-s + (0.524 + 1.26i)7-s + (0.554 − 0.831i)8-s + (−0.930 − 0.365i)9-s + (−0.0427 + 0.159i)10-s + (−1.21 + 0.242i)11-s + (−0.450 − 0.892i)12-s + (−0.496 + 0.331i)13-s + (0.903 + 1.02i)14-s + (0.118 + 0.115i)15-s + (0.257 − 0.966i)16-s + (1.16 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73380 - 1.02663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73380 - 1.02663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 + 0.454i)T \) |
| 3 | \( 1 + (-0.322 + 1.70i)T \) |
| good | 5 | \( 1 + (0.205 - 0.307i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.38 - 3.34i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.03 - 0.802i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.78 - 1.19i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.80 + 4.80i)T - 17iT^{2} \) |
| 19 | \( 1 + (4.56 - 3.05i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (0.318 - 0.769i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (0.933 - 4.69i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 + (-5.93 + 8.88i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (2.81 + 1.16i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.497 + 0.0989i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (3.50 + 3.50i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.878 - 4.41i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (7.88 + 5.27i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.65 + 8.31i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.680 - 0.135i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (7.27 - 3.01i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (6.01 + 2.49i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-6.21 - 6.21i)T + 79iT^{2} \) |
| 83 | \( 1 + (-1.80 - 2.70i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-4.83 + 2.00i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−12.39556625177491395875188087989, −11.81405587629781479074239173023, −10.80302967452064396470122315537, −9.419392599113818552271125662674, −8.036283917455981369674327654444, −7.16169625488202192776851627020, −5.80569004459584738812704657165, −5.06036712173002681543864833817, −3.01094701712217556876134568220, −2.02612774597495383510207150508,
2.84028191028812425500649940225, 4.20026411068760715702691987256, 4.89612195145673213266241628551, 6.17310874901103197521899094295, 7.79082923741771865712683721081, 8.256629892764085295118987280052, 10.27811545192288565775805904994, 10.56689205803971688516784519572, 11.72619016956558020541202451404, 12.98983017690510165845760313840
