L(s) = 1 | + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.813 + 0.469i)5-s + (0.866 + 0.5i)6-s + (2.42 + 1.06i)7-s − i·8-s + (−0.499 − 0.866i)9-s + (−0.469 + 0.813i)10-s + (1.02 + 0.592i)11-s + (−0.5 + 0.866i)12-s + (−2.78 + 2.28i)13-s + (−1.06 + 2.42i)14-s + (0.813 − 0.469i)15-s + 16-s + 4.44·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.288 − 0.499i)3-s − 0.5·4-s + (0.363 + 0.210i)5-s + (0.353 + 0.204i)6-s + (0.915 + 0.402i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.148 + 0.257i)10-s + (0.309 + 0.178i)11-s + (−0.144 + 0.249i)12-s + (−0.772 + 0.634i)13-s + (−0.284 + 0.647i)14-s + (0.210 − 0.121i)15-s + 0.250·16-s + 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61295 + 0.726976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61295 + 0.726976i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.42 - 1.06i)T \) |
| 13 | \( 1 + (2.78 - 2.28i)T \) |
good | 5 | \( 1 + (-0.813 - 0.469i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.02 - 0.592i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + (-2.19 + 1.26i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + (-4.86 - 8.41i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.13 + 0.656i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.99iT - 37T^{2} \) |
| 41 | \( 1 + (3.93 - 2.27i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.49 - 6.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.13 - 2.96i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.41 - 4.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.56iT - 59T^{2} \) |
| 61 | \( 1 + (6.79 + 11.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.0 + 6.38i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.59 + 3.80i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.28 + 4.20i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.88 - 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 + 14.7iT - 89T^{2} \) |
| 97 | \( 1 + (11.3 + 6.52i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.91483298284831067535795078822, −9.763729978115947101659187866432, −9.035810474171602936933561494493, −8.097963270038844494305787222022, −7.34410175944933951420670395657, −6.50247924230873438379195284261, −5.42419332364981768925528807933, −4.55626652881492371195514551418, −2.94917551639980725287890159655, −1.53226803201635069116792718134,
1.25131761854073082846487029069, 2.71023934173279386951431837579, 3.86778896490085176865597712533, 4.92962146135346057485998126606, 5.66645027303017340217509983583, 7.37182011731021728483766650700, 8.173910811508805100010347453845, 9.068028538477221172803340321424, 10.14903481575564989657311435971, 10.32816265377722861315553139226