L(s) = 1 | − 0.889i·2-s + (0.351 + 1.69i)3-s + 1.20·4-s + 1.21·5-s + (1.50 − 0.312i)6-s + (1.43 − 2.22i)7-s − 2.85i·8-s + (−2.75 + 1.19i)9-s − 1.08i·10-s − 5.85i·11-s + (0.424 + 2.05i)12-s + 5.22i·13-s + (−1.97 − 1.27i)14-s + (0.426 + 2.06i)15-s − 0.117·16-s + 17-s + ⋯ |
L(s) = 1 | − 0.628i·2-s + (0.202 + 0.979i)3-s + 0.604·4-s + 0.543·5-s + (0.615 − 0.127i)6-s + (0.543 − 0.839i)7-s − 1.00i·8-s + (−0.917 + 0.397i)9-s − 0.341i·10-s − 1.76i·11-s + (0.122 + 0.592i)12-s + 1.44i·13-s + (−0.527 − 0.341i)14-s + (0.110 + 0.532i)15-s − 0.0294·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81851 - 0.340151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81851 - 0.340151i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.351 - 1.69i)T \) |
| 7 | \( 1 + (-1.43 + 2.22i)T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 0.889iT - 2T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 11 | \( 1 + 5.85iT - 11T^{2} \) |
| 13 | \( 1 - 5.22iT - 13T^{2} \) |
| 19 | \( 1 - 0.437iT - 19T^{2} \) |
| 23 | \( 1 - 7.36iT - 23T^{2} \) |
| 29 | \( 1 - 3.77iT - 29T^{2} \) |
| 31 | \( 1 - 1.75iT - 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 + 0.530iT - 53T^{2} \) |
| 59 | \( 1 - 0.268T + 59T^{2} \) |
| 61 | \( 1 - 7.68iT - 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 4.66iT - 71T^{2} \) |
| 73 | \( 1 + 6.01iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 9.27T + 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−11.36730384612176801753830726449, −10.57921602837862832401844047032, −9.781746912379781519153941415729, −8.906878861305230408235356620319, −7.75589607989149419973724512947, −6.47687900255299750868826878646, −5.43103113815979202898257091489, −4.00859513218894836975051826761, −3.19822132852735495995397829667, −1.61316102996726409399010165007,
1.90430397492364927671975600999, 2.65842882262567949445701896186, 5.00554798765920718012372060662, 5.91273039969818768694313718367, 6.72460993780559949323154360296, 7.79265023433024120731353204400, 8.227384858583128412830029179890, 9.555513109671742443735937541661, 10.61875912725019068950564799445, 11.81417904240923804281867187887