| L(s) = 1 | + (−2.54 + 0.344i)2-s + (4.41 − 1.21i)4-s + (−0.225 − 0.310i)5-s + (3.10 + 2.96i)7-s + (−6.07 + 2.59i)8-s + (0.680 + 0.711i)10-s + (5.26 + 0.474i)11-s + (0.0382 + 0.424i)13-s + (−8.89 − 6.46i)14-s + (6.68 − 3.99i)16-s + (0.912 + 2.80i)17-s + (−6.30 + 2.36i)19-s + (−1.37 − 1.09i)20-s + (−13.5 + 0.608i)22-s + (5.74 + 2.76i)23-s + ⋯ |
| L(s) = 1 | + (−1.79 + 0.243i)2-s + (2.20 − 0.608i)4-s + (−0.100 − 0.138i)5-s + (1.17 + 1.12i)7-s + (−2.14 + 0.917i)8-s + (0.215 + 0.224i)10-s + (1.58 + 0.142i)11-s + (0.0106 + 0.117i)13-s + (−2.37 − 1.72i)14-s + (1.67 − 0.998i)16-s + (0.221 + 0.680i)17-s + (−1.44 + 0.542i)19-s + (−0.307 − 0.244i)20-s + (−2.88 + 0.129i)22-s + (1.19 + 0.576i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.643692 + 0.410619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643692 + 0.410619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (-1.22 - 8.33i)T \) |
| good | 2 | \( 1 + (2.54 - 0.344i)T + (1.92 - 0.532i)T^{2} \) |
| 5 | \( 1 + (0.225 + 0.310i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 2.96i)T + (0.314 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-5.26 - 0.474i)T + (10.8 + 1.96i)T^{2} \) |
| 13 | \( 1 + (-0.0382 - 0.424i)T + (-12.7 + 2.32i)T^{2} \) |
| 17 | \( 1 + (-0.912 - 2.80i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (6.30 - 2.36i)T + (14.3 - 12.5i)T^{2} \) |
| 23 | \( 1 + (-5.74 - 2.76i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (9.88 + 0.444i)T + (28.8 + 2.59i)T^{2} \) |
| 31 | \( 1 + (-3.38 + 5.65i)T + (-14.6 - 27.2i)T^{2} \) |
| 37 | \( 1 + (0.0838 - 0.0403i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (0.878 - 3.84i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.27 + 2.37i)T + (-23.6 - 35.8i)T^{2} \) |
| 47 | \( 1 + (-0.643 + 0.974i)T + (-18.4 - 43.2i)T^{2} \) |
| 53 | \( 1 + (-0.448 - 0.123i)T + (45.4 + 27.1i)T^{2} \) |
| 59 | \( 1 + (-3.78 - 3.30i)T + (7.91 + 58.4i)T^{2} \) |
| 61 | \( 1 + (5.10 - 4.88i)T + (2.73 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-2.07 - 7.51i)T + (-57.5 + 34.3i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 9.28i)T + (-70.3 + 19.4i)T^{2} \) |
| 79 | \( 1 + (-0.405 - 0.947i)T + (-54.5 + 57.1i)T^{2} \) |
| 83 | \( 1 + (-3.50 + 4.01i)T + (-11.1 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-4.40 + 15.9i)T + (-76.4 - 45.6i)T^{2} \) |
| 97 | \( 1 + (-1.53 + 0.350i)T + (87.3 - 42.0i)T^{2} \) |
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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
−10.59999906574537337705991512701, −9.567721297270298184361321494310, −8.820923958349975303566528766152, −8.447545104796974857065963730026, −7.52940922585431704976720836617, −6.48889255080592081613854066459, −5.71155702215610852933459404926, −4.17744574066327437448301042929, −2.22720301278495064463109429933, −1.38275865293924575019098895103,
0.860772901564593034070413612333, 1.86265931593378796224667741348, 3.49136619577927616926525739584, 4.76963448825679438956593972358, 6.57702874645891025119889791215, 7.13249045572492264471191748709, 7.914962737944767992075600403424, 8.886937370468662910181535495535, 9.311500210751197899912635139607, 10.54872743784411556239769548483
