L(s) = 1 | + (0.951 − 2.66i)2-s + (−6.19 − 5.06i)4-s − 10.8i·5-s + (−15.1 − 10.6i)7-s + (−19.3 + 11.6i)8-s + (−28.8 − 10.3i)10-s + 45.0i·11-s − 40.3i·13-s + (−42.8 + 30.1i)14-s + (12.6 + 62.7i)16-s + 8.11i·17-s − 53.3·19-s + (−54.9 + 67.1i)20-s + (119. + 42.8i)22-s + 55.7i·23-s + ⋯ |
L(s) = 1 | + (0.336 − 0.941i)2-s + (−0.773 − 0.633i)4-s − 0.970i·5-s + (−0.817 − 0.576i)7-s + (−0.856 + 0.515i)8-s + (−0.913 − 0.326i)10-s + 1.23i·11-s − 0.861i·13-s + (−0.817 + 0.575i)14-s + (0.197 + 0.980i)16-s + 0.115i·17-s − 0.643·19-s + (−0.614 + 0.750i)20-s + (1.16 + 0.415i)22-s + 0.505i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4885942994\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4885942994\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 2.66i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (15.1 + 10.6i)T \) |
good | 5 | \( 1 + 10.8iT - 125T^{2} \) |
| 11 | \( 1 - 45.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 40.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 8.11iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 53.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 55.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 354.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 42.7iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 23.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 437.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 649.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 916. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 736. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 600. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 673. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 585. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 920. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 638. iT - 9.12e5T^{2} \) |
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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
−10.84677757955111882754615140536, −10.03359205256711011727937862757, −9.306597305466872540150590391814, −8.226405649666813029758281426857, −6.79162098659087742881528389988, −5.37444573187218228748386099702, −4.46226540784416266957681415399, −3.29955416054073761995044520666, −1.65257225838441165855640533742, −0.17013813352285516130640906202,
2.81131174362307648565906204323, 3.80825539119958056317649513540, 5.39806985034113767291618172465, 6.46279500200137004810695115178, 6.91269084318761826726684641760, 8.397390272238430774576338389765, 9.090089411634967730803950817398, 10.30093616222840185098582049453, 11.41350094077903719463479204815, 12.44561343758247341714748811353