| L(s) = 1 | + (−0.359 − 2.27i)2-s + (0.374 − 1.69i)3-s + (−3.12 + 1.01i)4-s + (0.136 + 0.0697i)5-s + (−3.97 − 0.243i)6-s + (1.38 − 0.333i)7-s + (1.34 + 2.63i)8-s + (−2.71 − 1.26i)9-s + (0.109 − 0.335i)10-s + (−0.659 + 0.772i)11-s + (0.545 + 5.66i)12-s + (2.99 + 4.88i)13-s + (−1.25 − 3.03i)14-s + (0.169 − 0.205i)15-s + (0.189 − 0.137i)16-s + (4.60 − 0.362i)17-s + ⋯ |
| L(s) = 1 | + (−0.254 − 1.60i)2-s + (0.216 − 0.976i)3-s + (−1.56 + 0.507i)4-s + (0.0612 + 0.0311i)5-s + (−1.62 − 0.0992i)6-s + (0.524 − 0.125i)7-s + (0.475 + 0.932i)8-s + (−0.906 − 0.422i)9-s + (0.0345 − 0.106i)10-s + (−0.198 + 0.232i)11-s + (0.157 + 1.63i)12-s + (0.830 + 1.35i)13-s + (−0.335 − 0.810i)14-s + (0.0436 − 0.0530i)15-s + (0.0472 − 0.0343i)16-s + (1.11 − 0.0879i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.101643 - 0.961947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101643 - 0.961947i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.374 + 1.69i)T \) |
| 41 | \( 1 + (0.530 - 6.38i)T \) |
| good | 2 | \( 1 + (0.359 + 2.27i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (-0.136 - 0.0697i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 0.333i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (0.659 - 0.772i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 4.88i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-4.60 + 0.362i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-2.40 + 3.91i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (5.59 + 4.06i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0606 - 0.00477i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-4.62 - 1.50i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.402 - 1.23i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-8.28 + 1.31i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (1.46 - 6.08i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.764 - 9.70i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-0.765 + 1.05i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.108 - 0.686i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (6.60 + 7.73i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-10.1 - 8.65i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (9.51 - 9.51i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.107 - 0.258i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 5.23iT - 83T^{2} \) |
| 89 | \( 1 + (3.19 + 13.2i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (2.64 - 2.25i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.62845912160515938508285334554, −11.84911394849954185707826673500, −11.20100220594503669624237891575, −9.945993675159610121076146878143, −8.877627614294127544053754237144, −7.84195201765213283041434047445, −6.32541468674860863481641153635, −4.31261076885100842517545366444, −2.70577033593662845840926950775, −1.36416454041728511933984304111,
3.64912548253830699809064611338, 5.38654359030107289829388677677, 5.80492776599680481350053570188, 7.84221085953722991566872332511, 8.173298675032582422052212102382, 9.480536820015884520293525412079, 10.39926156813881119407266624149, 11.74746921451163738194309627194, 13.49558684088294582664007323114, 14.28401606699359184563399575580
