| L(s) = 1 | + 2.33·3-s + 3.10·5-s − 3.54·9-s + 4.69i·11-s − 6.88·13-s + 7.24·15-s + 16.9i·17-s − 26.2·19-s − 25.8·23-s − 15.3·25-s − 29.3·27-s − 42.2i·29-s − 18.3i·31-s + 10.9i·33-s − 49.8i·37-s + ⋯ |
| L(s) = 1 | + 0.778·3-s + 0.620·5-s − 0.393·9-s + 0.426i·11-s − 0.529·13-s + 0.482·15-s + 0.999i·17-s − 1.37·19-s − 1.12·23-s − 0.615·25-s − 1.08·27-s − 1.45i·29-s − 0.592i·31-s + 0.332i·33-s − 1.34i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01785169058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01785169058\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.33T + 9T^{2} \) |
| 5 | \( 1 - 3.10T + 25T^{2} \) |
| 11 | \( 1 - 4.69iT - 121T^{2} \) |
| 13 | \( 1 + 6.88T + 169T^{2} \) |
| 17 | \( 1 - 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 26.2T + 361T^{2} \) |
| 23 | \( 1 + 25.8T + 529T^{2} \) |
| 29 | \( 1 + 42.2iT - 841T^{2} \) |
| 31 | \( 1 + 18.3iT - 961T^{2} \) |
| 37 | \( 1 + 49.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 106.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 93.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 13.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 2.15T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 88.9iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.635604055749440731344233153840, −8.853290845989855907774558910648, −8.092997473611036169233447137826, −7.50325896986443774232359594263, −6.16449187634847347851464891884, −5.86577910073583291640656584731, −4.45908748567358834633372953291, −3.74975903479968207678457851917, −2.36269216470906854204226415195, −1.99549313084865646330086794311,
0.00369312642214078743621022791, 1.75956644494727168471980909237, 2.63182209218693010418263909575, 3.47942794281963590234951432318, 4.64647583118711907716378270732, 5.57627304578997018954547762163, 6.39285623736960798050227153475, 7.29605300981723170659144226019, 8.271421882616165205983832021876, 8.744611583017028643998915788387
