L(s) = 1 | + (0.305 − 1.97i)2-s − 1.73·3-s + (−3.81 − 1.20i)4-s + (−0.529 + 3.42i)6-s − 0.329·7-s + (−3.55 + 7.16i)8-s + 2.99·9-s + 20.4i·11-s + (6.60 + 2.09i)12-s + 0.416i·13-s + (−0.100 + 0.652i)14-s + (13.0 + 9.21i)16-s − 18.5i·17-s + (0.917 − 5.92i)18-s + 12.4i·19-s + ⋯ |
L(s) = 1 | + (0.152 − 0.988i)2-s − 0.577·3-s + (−0.953 − 0.302i)4-s + (−0.0882 + 0.570i)6-s − 0.0471·7-s + (−0.444 + 0.895i)8-s + 0.333·9-s + 1.86i·11-s + (0.550 + 0.174i)12-s + 0.0320i·13-s + (−0.00720 + 0.0465i)14-s + (0.817 + 0.575i)16-s − 1.09i·17-s + (0.0509 − 0.329i)18-s + 0.655i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04004 + 0.0816926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04004 + 0.0816926i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.305 + 1.97i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.329T + 49T^{2} \) |
| 11 | \( 1 - 20.4iT - 121T^{2} \) |
| 13 | \( 1 - 0.416iT - 169T^{2} \) |
| 17 | \( 1 + 18.5iT - 289T^{2} \) |
| 19 | \( 1 - 12.4iT - 361T^{2} \) |
| 23 | \( 1 - 23.2T + 529T^{2} \) |
| 29 | \( 1 - 23.9T + 841T^{2} \) |
| 31 | \( 1 - 42.0iT - 961T^{2} \) |
| 37 | \( 1 - 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 55.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 81.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 72.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.88T + 6.88e3T^{2} \) |
| 89 | \( 1 - 61.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 95.5iT - 9.40e3T^{2} \) |
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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
−11.67130903484004597853763079235, −10.62897020904518119067564195961, −9.882346991653018688400703595182, −9.129113018777297631582722998552, −7.68594554024682729195846018376, −6.56725366058339207483265095010, −5.07380625833422574777604223741, −4.49068605567045411696448670273, −2.88231028383489034626104474656, −1.38846097723471396251476739114,
0.58471224703899958163219922539, 3.31402135817488872994881985220, 4.57207768593762260009211420871, 5.81452251009822759922492454308, 6.29309808362402542562583325566, 7.56723017082575819213827420784, 8.515918989913406885570968553181, 9.352855142765151035846818179918, 10.68789879877308352829036402929, 11.40637598155423399761021325796