L(s) = 1 | − 0.542·2-s + (−1.58 − 0.693i)3-s − 1.70·4-s + i·5-s + (0.861 + 0.376i)6-s − i·7-s + 2.01·8-s + (2.03 + 2.20i)9-s − 0.542i·10-s + (−2.53 − 2.14i)11-s + (2.70 + 1.18i)12-s + 0.789i·13-s + 0.542i·14-s + (0.693 − 1.58i)15-s + 2.31·16-s − 0.309·17-s + ⋯ |
L(s) = 1 | − 0.383·2-s + (−0.916 − 0.400i)3-s − 0.852·4-s + 0.447i·5-s + (0.351 + 0.153i)6-s − 0.377i·7-s + 0.710·8-s + (0.679 + 0.733i)9-s − 0.171i·10-s + (−0.763 − 0.645i)11-s + (0.781 + 0.341i)12-s + 0.219i·13-s + 0.145i·14-s + (0.179 − 0.409i)15-s + 0.579·16-s − 0.0750·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2001772462\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2001772462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.58 + 0.693i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (2.53 + 2.14i)T \) |
good | 2 | \( 1 + 0.542T + 2T^{2} \) |
| 13 | \( 1 - 0.789iT - 13T^{2} \) |
| 17 | \( 1 + 0.309T + 17T^{2} \) |
| 19 | \( 1 + 4.40iT - 19T^{2} \) |
| 23 | \( 1 + 2.63iT - 23T^{2} \) |
| 29 | \( 1 - 8.52T + 29T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 - 9.93iT - 43T^{2} \) |
| 47 | \( 1 + 2.67iT - 47T^{2} \) |
| 53 | \( 1 - 4.44iT - 53T^{2} \) |
| 59 | \( 1 - 9.09iT - 59T^{2} \) |
| 61 | \( 1 - 7.24iT - 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.04iT - 71T^{2} \) |
| 73 | \( 1 - 5.23iT - 73T^{2} \) |
| 79 | \( 1 - 7.41iT - 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−10.32587712160318695414146145169, −9.252887529138275463680341111717, −8.381084285507624141818937672309, −7.55753621182379638635078105771, −6.82410927283546059308406734433, −5.84197685999878434856459776416, −4.96064539989101058631931293598, −4.18495066348976793554085221158, −2.74582302632775479757651421605, −1.09126488183612804797582708821,
0.14090743617052091655717573315, 1.65104756873099516276441449217, 3.52644229852823752863161859867, 4.53754414929213985831730099054, 5.21610582805841669530754658851, 5.86065425451757972542257040118, 7.13920643251920066729264322090, 8.012232722397705135474360929658, 8.852202085693328028583116358234, 9.629563527065338597450427862652