| L(s) = 1 | + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.499 − 3.15i)5-s + (0.671 + 0.786i)7-s + (−0.987 − 0.156i)8-s + (2.58 − 1.87i)10-s + (−0.100 + 0.164i)11-s + (−0.239 − 3.03i)13-s + (−0.395 + 0.955i)14-s + (−0.309 − 0.951i)16-s + (−0.422 − 1.76i)17-s + (−0.00761 + 0.0966i)19-s + (2.84 + 1.44i)20-s + (−0.192 − 0.0151i)22-s + (2.70 − 8.32i)23-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (−0.293 + 0.404i)4-s + (−0.223 − 1.40i)5-s + (0.253 + 0.297i)7-s + (−0.349 − 0.0553i)8-s + (0.816 − 0.593i)10-s + (−0.0304 + 0.0496i)11-s + (−0.0663 − 0.842i)13-s + (−0.105 + 0.255i)14-s + (−0.0772 − 0.237i)16-s + (−0.102 − 0.427i)17-s + (−0.00174 + 0.0221i)19-s + (0.635 + 0.324i)20-s + (−0.0410 − 0.00323i)22-s + (0.564 − 1.73i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40494 - 0.551647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40494 - 0.551647i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 - 0.891i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (0.788 + 6.35i)T \) |
| good | 5 | \( 1 + (0.499 + 3.15i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-0.671 - 0.786i)T + (-1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (0.100 - 0.164i)T + (-4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (0.239 + 3.03i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (0.422 + 1.76i)T + (-15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (0.00761 - 0.0966i)T + (-18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (-2.70 + 8.32i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.74 + 7.27i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (-1.03 - 1.42i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.06 - 2.22i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (3.32 - 1.69i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-7.60 - 6.49i)T + (7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (6.19 + 1.48i)T + (47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (2.72 + 0.886i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.34 - 6.57i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-1.53 - 2.50i)T + (-30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (-7.19 - 4.41i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (-4.69 - 4.69i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.03 + 2.49i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 11.5iT - 83T^{2} \) |
| 89 | \( 1 + (11.9 - 10.2i)T + (13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (-0.141 + 0.0864i)T + (44.0 - 86.4i)T^{2} \) |
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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
−10.11845820973768220140929719246, −9.130832668559565917246766334345, −8.413506675423432916773363111102, −7.897747253954823469107031792096, −6.70398720516817879259226794911, −5.63607270004097209083245120122, −4.89001590995526240031256956821, −4.18236677698956915999413135646, −2.63604497399337868805383254245, −0.73190556683607622312096113949,
1.68533870339572999696152181096, 2.98731398996539727060683787872, 3.75431353553053542629109982825, 4.87822558909848585143179162477, 6.10259851609539499911878324309, 6.97054136788366675311881772858, 7.71559183281484407844215405800, 8.989484145881901094638166481464, 9.838614493289858729026775085870, 10.75187874824711207755629054694
