L(s) = 1 | + 0.630i·3-s + 3.98·7-s + 8.60·9-s − 5.54·11-s + 23.1i·13-s − 16.5·17-s + (11.6 + 15.0i)19-s + 2.51i·21-s − 6.86·23-s + 11.0i·27-s − 37.0i·29-s − 11.1i·31-s − 3.49i·33-s + 66.6i·37-s − 14.5·39-s + ⋯ |
L(s) = 1 | + 0.210i·3-s + 0.569·7-s + 0.955·9-s − 0.504·11-s + 1.77i·13-s − 0.970·17-s + (0.613 + 0.790i)19-s + 0.119i·21-s − 0.298·23-s + 0.410i·27-s − 1.27i·29-s − 0.358i·31-s − 0.105i·33-s + 1.80i·37-s − 0.373·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.490629480\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.490629480\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-11.6 - 15.0i)T \) |
good | 3 | \( 1 - 0.630iT - 9T^{2} \) |
| 7 | \( 1 - 3.98T + 49T^{2} \) |
| 11 | \( 1 + 5.54T + 121T^{2} \) |
| 13 | \( 1 - 23.1iT - 169T^{2} \) |
| 17 | \( 1 + 16.5T + 289T^{2} \) |
| 23 | \( 1 + 6.86T + 529T^{2} \) |
| 29 | \( 1 + 37.0iT - 841T^{2} \) |
| 31 | \( 1 + 11.1iT - 961T^{2} \) |
| 37 | \( 1 - 66.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 129.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 66.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.0iT - 9.40e3T^{2} \) |
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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
−9.592765877120268854154583419741, −8.410508139007485606924758731540, −7.86165333023924527360047836861, −6.84342969083432647502251557107, −6.32762984703669409134274192290, −5.00039761705662766062588601306, −4.48460723350183078772458360552, −3.65464589476509515754156414423, −2.20912981020746269994436537356, −1.44314171200655227270169908555,
0.37398247647206728823353445724, 1.57404680223738504055777917030, 2.69021600969195035803814488642, 3.69490740195179558207791593746, 4.88067462587708987101144603904, 5.31052405959850723045603817762, 6.47524955430377874599744901281, 7.32024007704375864404892739223, 7.86292089450175076008318374224, 8.665568003676176775896412079326