L(s) = 1 | + (0.963 − 0.556i)2-s + (1.73 + 0.0368i)3-s + (−0.381 + 0.660i)4-s + (−0.5 + 0.866i)5-s + (1.68 − 0.927i)6-s + (2.64 − 0.0472i)7-s + 3.07i·8-s + (2.99 + 0.127i)9-s + 1.11i·10-s + (−2.20 + 1.27i)11-s + (−0.684 + 1.12i)12-s + (−5.31 − 3.07i)13-s + (2.52 − 1.51i)14-s + (−0.897 + 1.48i)15-s + (0.946 + 1.63i)16-s + 3.62·17-s + ⋯ |
L(s) = 1 | + (0.681 − 0.393i)2-s + (0.999 + 0.0212i)3-s + (−0.190 + 0.330i)4-s + (−0.223 + 0.387i)5-s + (0.689 − 0.378i)6-s + (0.999 − 0.0178i)7-s + 1.08i·8-s + (0.999 + 0.0424i)9-s + 0.351i·10-s + (−0.663 + 0.383i)11-s + (−0.197 + 0.326i)12-s + (−1.47 − 0.851i)13-s + (0.674 − 0.405i)14-s + (−0.231 + 0.382i)15-s + (0.236 + 0.409i)16-s + 0.878·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.32011 + 0.174211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.32011 + 0.174211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.73 - 0.0368i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.64 + 0.0472i)T \) |
good | 2 | \( 1 + (-0.963 + 0.556i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.20 - 1.27i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.31 + 3.07i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 5.85iT - 19T^{2} \) |
| 23 | \( 1 + (0.351 + 0.202i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.74 + 4.47i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.10 + 1.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + (2.20 - 3.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.15 + 1.99i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.93 + 3.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 + (5.15 - 8.93i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.26 - 2.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.48 + 7.76i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.50iT - 71T^{2} \) |
| 73 | \( 1 + 2.90iT - 73T^{2} \) |
| 79 | \( 1 + (-5.76 - 9.99i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.99 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + (-9.04 + 5.22i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.03888508007023435990486633334, −10.77430480926364868983164034632, −9.944728806546103268440547037693, −8.661493721276018921284730137271, −7.82971181539892748115274828462, −7.28784088586567410176848853058, −5.14880224986594499675074448922, −4.52275334658290208941785424207, −3.11062837686081423151184802039, −2.34657313356448224409400157057,
1.70363194604061034421708260539, 3.47283314756509516435015129898, 4.68276561543940068161758052172, 5.30123267120128802376868177276, 6.89768754762605676893450839187, 7.82895528761531924765418480337, 8.648474848369951552702975183767, 9.767854336962016382889601279802, 10.45533823871311751227780670335, 12.08042387496424890227286600854