| L(s) = 1 | + (−1.73 + 0.344i)3-s + (−0.638 + 0.955i)5-s + (−5.52 − 8.26i)7-s + (−5.43 + 2.25i)9-s + (2.46 − 12.3i)11-s + (−1.27 + 1.27i)13-s + (0.776 − 1.87i)15-s + (−9.29 − 14.2i)17-s + (−23.4 − 9.70i)19-s + (12.4 + 12.4i)21-s + (−14.9 − 2.97i)23-s + (9.06 + 21.8i)25-s + (21.8 − 14.5i)27-s + (2.50 + 1.67i)29-s + (5.93 + 29.8i)31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.114i)3-s + (−0.127 + 0.191i)5-s + (−0.788 − 1.18i)7-s + (−0.603 + 0.250i)9-s + (0.223 − 1.12i)11-s + (−0.0978 + 0.0978i)13-s + (0.0517 − 0.125i)15-s + (−0.547 − 0.837i)17-s + (−1.23 − 0.510i)19-s + (0.591 + 0.591i)21-s + (−0.649 − 0.129i)23-s + (0.362 + 0.875i)25-s + (0.809 − 0.540i)27-s + (0.0863 + 0.0576i)29-s + (0.191 + 0.961i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.178696 - 0.431502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178696 - 0.431502i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (9.29 + 14.2i)T \) |
| good | 3 | \( 1 + (1.73 - 0.344i)T + (8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (0.638 - 0.955i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (5.52 + 8.26i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.46 + 12.3i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (1.27 - 1.27i)T - 169iT^{2} \) |
| 19 | \( 1 + (23.4 + 9.70i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (14.9 + 2.97i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-2.50 - 1.67i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-5.93 - 29.8i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-42.7 + 8.49i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-9.73 - 14.5i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-27.2 + 11.2i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (30.8 - 30.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (67.9 + 28.1i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (7.60 + 18.3i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-21.2 + 14.2i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 96.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (39.9 - 7.95i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-51.8 + 77.5i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (22.6 - 113. i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (22.1 - 53.4i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (94.0 + 94.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (117. + 78.1i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.64767588387800765626401766439, −11.16906012189251172229401275850, −10.91833913552285407953205666588, −9.593237748708685154229197370962, −8.355989669247345187301627611070, −6.94804095207605196901519633578, −6.08663310556556474524073107713, −4.55063006977162455742880182429, −3.11056654088555049515491260217, −0.31662048414165728852496345049,
2.40157293783689947006346276638, 4.30193609683066712820250346728, 5.86577157337478545613056207580, 6.49747883890051672336254542762, 8.203624817266228597703790290536, 9.195424062245407995321949501703, 10.26460429840872729529971784115, 11.56076839364715683358758564228, 12.41024667454575717036001207380, 12.88514879101749013228198360282
