L(s) = 1 | + (−0.707 − 1.87i)2-s + (−1.41 − 2.64i)3-s + (−3 + 2.64i)4-s + 5.65·5-s + (−3.94 + 4.51i)6-s + 4·7-s + (7.07 + 3.74i)8-s + (−5 + 7.48i)9-s + (−4.00 − 10.5i)10-s − 8.48·11-s + (11.2 + 4.19i)12-s + 10.5i·13-s + (−2.82 − 7.48i)14-s + (−8.00 − 14.9i)15-s + (1.99 − 15.8i)16-s − 14.9i·17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.935i)2-s + (−0.471 − 0.881i)3-s + (−0.750 + 0.661i)4-s + 1.13·5-s + (−0.658 + 0.752i)6-s + 0.571·7-s + (0.883 + 0.467i)8-s + (−0.555 + 0.831i)9-s + (−0.400 − 1.05i)10-s − 0.771·11-s + (0.936 + 0.349i)12-s + 0.814i·13-s + (−0.202 − 0.534i)14-s + (−0.533 − 0.997i)15-s + (0.124 − 0.992i)16-s − 0.880i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00418 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00418 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.534844 - 0.532608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534844 - 0.532608i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.87i)T \) |
| 3 | \( 1 + (1.41 + 2.64i)T \) |
good | 5 | \( 1 - 5.65T + 25T^{2} \) |
| 7 | \( 1 - 4T + 49T^{2} \) |
| 11 | \( 1 + 8.48T + 121T^{2} \) |
| 13 | \( 1 - 10.5iT - 169T^{2} \) |
| 17 | \( 1 + 14.9iT - 289T^{2} \) |
| 19 | \( 1 - 5.29iT - 361T^{2} \) |
| 23 | \( 1 - 29.9iT - 529T^{2} \) |
| 29 | \( 1 + 16.9T + 841T^{2} \) |
| 31 | \( 1 + 4T + 961T^{2} \) |
| 37 | \( 1 + 52.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 29.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.29iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 48.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 95.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 89.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 124T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.82T + 6.88e3T^{2} \) |
| 89 | \( 1 - 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 118T + 9.40e3T^{2} \) |
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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
−17.73030614622155761934021313832, −16.59151766647779599834726882434, −14.01171933621875929984920208097, −13.30585491727059322223027458989, −11.94041167474267908976916677818, −10.79036843865912096372979466727, −9.292175672440491200167963865656, −7.55293857840674426208737803386, −5.38400542037203543276090817067, −1.99519762672624888677905786603,
4.98220427408142433639761568124, 6.14447145950804349033210209199, 8.351408340978118145097745896039, 9.846670408141429861209173533844, 10.72062752973857449709958500144, 13.06787330122284560108302223006, 14.49757130445901706271734898481, 15.42479459457495621408542591918, 16.75613951283113103331764162205, 17.55015296633442841629898496836