L(s) = 1 | + (1.41 + 0.0402i)2-s + (1.99 + 0.113i)4-s + (1.51 + 0.626i)5-s + (−1.32 − 1.32i)7-s + (2.81 + 0.241i)8-s + (2.11 + 0.946i)10-s + (0.938 − 2.26i)11-s + (−4.73 + 1.96i)13-s + (−1.82 − 1.92i)14-s + (3.97 + 0.454i)16-s + 5.78i·17-s + (−1.04 + 0.431i)19-s + (2.94 + 1.42i)20-s + (1.41 − 3.16i)22-s + (4.29 − 4.29i)23-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0284i)2-s + (0.998 + 0.0568i)4-s + (0.676 + 0.280i)5-s + (−0.500 − 0.500i)7-s + (0.996 + 0.0852i)8-s + (0.668 + 0.299i)10-s + (0.283 − 0.683i)11-s + (−1.31 + 0.543i)13-s + (−0.486 − 0.514i)14-s + (0.993 + 0.113i)16-s + 1.40i·17-s + (−0.239 + 0.0990i)19-s + (0.659 + 0.318i)20-s + (0.302 − 0.675i)22-s + (0.895 − 0.895i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39236 + 0.0647935i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39236 + 0.0647935i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 - 0.0402i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.51 - 0.626i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (1.32 + 1.32i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.938 + 2.26i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (4.73 - 1.96i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 5.78iT - 17T^{2} \) |
| 19 | \( 1 + (1.04 - 0.431i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.29 + 4.29i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.389 + 0.940i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + (3.67 + 1.52i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.474 + 0.474i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.409 - 0.987i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 2.73iT - 47T^{2} \) |
| 53 | \( 1 + (4.55 - 10.9i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (8.68 + 3.59i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.48 - 3.58i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-6.11 - 14.7i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (10.5 + 10.5i)T + 71iT^{2} \) |
| 73 | \( 1 + (-6.86 + 6.86i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 + (-15.2 + 6.30i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.07 - 6.07i)T + 89iT^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−12.06816643118825624056207703040, −10.83182489931532126198266404382, −10.28434319303190266827222297468, −9.075231891889680999113658324467, −7.61306412184283134722752654707, −6.61046481187186936935462240843, −5.91138378238779425114833989673, −4.59576979009122131274169781359, −3.43094923767502322143328039397, −2.07041353149387267723044139409,
2.07397518715854664325664340959, 3.25357191494106228544427479867, 4.92923685192959067072655983915, 5.46533818366016352590732031046, 6.79062185436123555886743217178, 7.54151358548018374477223803001, 9.312605806929509625806774364570, 9.817991542222813413777285583732, 11.10313327288380196166873191247, 12.09847461876045157958051193555