| L(s) = 1 | + (−0.5 + 1.32i)2-s + (−1.50 − 1.32i)4-s − 1.73i·5-s + (2.50 − 1.32i)8-s + (2.29 + 0.866i)10-s − 2.64i·11-s − 3.46i·13-s + (0.500 + 3.96i)16-s + 6.92i·17-s + (−2.29 + 2.59i)20-s + (3.50 + 1.32i)22-s − 5.29i·23-s + 2.00·25-s + (4.58 + 1.73i)26-s − 5·29-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.935i)2-s + (−0.750 − 0.661i)4-s − 0.774i·5-s + (0.883 − 0.467i)8-s + (0.724 + 0.273i)10-s − 0.797i·11-s − 0.960i·13-s + (0.125 + 0.992i)16-s + 1.68i·17-s + (−0.512 + 0.580i)20-s + (0.746 + 0.282i)22-s − 1.10i·23-s + 0.400·25-s + (0.898 + 0.339i)26-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8200886341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8200886341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.5 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 9.16T + 47T^{2} \) |
| 53 | \( 1 + 7T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 7.93iT - 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.800870963372520952441849305030, −8.312759318337020633138409021045, −7.81940322769711816779573288865, −6.58356058048873490179702537438, −5.99289540515879073945625565177, −5.20182501429457144821064245584, −4.38328095269047818774587909125, −3.32524312924954652001708036831, −1.58464797345731596690516060120, −0.36522061699548891888520923289,
1.46131773783984180374372413399, 2.54510329369379419553273193999, 3.30426640783988546062337694758, 4.41061637432354085562968578428, 5.09983209461365445667346049628, 6.49331606650418778240929965600, 7.29381320173410052865729398359, 7.83099142584844082336514456237, 9.115330788611697096420605006482, 9.487249530717181084564735588106
