L(s) = 1 | + (0.909 + 0.909i)2-s + (0.442 + 0.442i)3-s − 2.34i·4-s + 4.16i·5-s + 0.805i·6-s − 9.68·7-s + (5.77 − 5.77i)8-s − 8.60i·9-s + (−3.78 + 3.78i)10-s + (−0.334 − 0.334i)11-s + (1.03 − 1.03i)12-s + 12.2i·13-s + (−8.81 − 8.81i)14-s + (−1.84 + 1.84i)15-s + 1.11·16-s + (6.80 + 6.80i)17-s + ⋯ |
L(s) = 1 | + (0.454 + 0.454i)2-s + (0.147 + 0.147i)3-s − 0.586i·4-s + 0.832i·5-s + 0.134i·6-s − 1.38·7-s + (0.721 − 0.721i)8-s − 0.956i·9-s + (−0.378 + 0.378i)10-s + (−0.0304 − 0.0304i)11-s + (0.0865 − 0.0865i)12-s + 0.940i·13-s + (−0.629 − 0.629i)14-s + (−0.122 + 0.122i)15-s + 0.0698·16-s + (0.400 + 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12415 + 0.251068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12415 + 0.251068i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-28.1 + 7.15i)T \) |
good | 2 | \( 1 + (-0.909 - 0.909i)T + 4iT^{2} \) |
| 3 | \( 1 + (-0.442 - 0.442i)T + 9iT^{2} \) |
| 5 | \( 1 - 4.16iT - 25T^{2} \) |
| 7 | \( 1 + 9.68T + 49T^{2} \) |
| 11 | \( 1 + (0.334 + 0.334i)T + 121iT^{2} \) |
| 13 | \( 1 - 12.2iT - 169T^{2} \) |
| 17 | \( 1 + (-6.80 - 6.80i)T + 289iT^{2} \) |
| 19 | \( 1 + (-14.6 - 14.6i)T + 361iT^{2} \) |
| 23 | \( 1 + 10.0T + 529T^{2} \) |
| 31 | \( 1 + (37.3 + 37.3i)T + 961iT^{2} \) |
| 37 | \( 1 + (45.0 - 45.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (-22.8 + 22.8i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (17.5 + 17.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-2.20 + 2.20i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 90.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (29.4 + 29.4i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + 31.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 99.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (3.96 - 3.96i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (40.3 + 40.3i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 - 137.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-83.1 - 83.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (79.9 - 79.9i)T - 9.40e3iT^{2} \) |
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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
−16.60221065080574347138383100082, −15.58867123480813835903575747945, −14.62615887862688524863274881452, −13.66482492402327738565023043578, −12.15498605986684551748281788420, −10.33678181107372981521418134657, −9.434726193625607864928453284584, −6.93979564666978871367798565554, −6.05667775407692816830534473862, −3.64788929232417560239361487827,
3.09957726086535191363110545545, 5.11422401381085990128893379332, 7.42575825059943410272127857067, 8.888286674034197731583425219879, 10.54336023713537912152418508851, 12.26516815167372795761522739183, 12.96424428251964812050542134412, 13.84994696833320518615562394800, 16.01868181820487541463993504476, 16.51646880103747692535453490435