| L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.394 − 2.49i)5-s + (−1.88 − 1.61i)7-s + (−0.987 + 0.156i)8-s + (−2.04 − 1.48i)10-s + (−3.63 + 2.22i)11-s + (−5.05 − 0.398i)13-s + (−2.29 + 0.949i)14-s + (−0.309 + 0.951i)16-s + (5.49 + 1.32i)17-s + (−4.35 + 0.342i)19-s + (−2.24 + 1.14i)20-s + (0.334 + 4.24i)22-s + (0.132 + 0.408i)23-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (−0.293 − 0.404i)4-s + (0.176 − 1.11i)5-s + (−0.713 − 0.609i)7-s + (−0.349 + 0.0553i)8-s + (−0.645 − 0.469i)10-s + (−1.09 + 0.670i)11-s + (−1.40 − 0.110i)13-s + (−0.612 + 0.253i)14-s + (−0.0772 + 0.237i)16-s + (1.33 + 0.320i)17-s + (−0.999 + 0.0786i)19-s + (−0.502 + 0.256i)20-s + (0.0712 + 0.905i)22-s + (0.0277 + 0.0852i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.154958 + 0.765656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.154958 + 0.765656i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.12 - 4.89i)T \) |
| good | 5 | \( 1 + (-0.394 + 2.49i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (1.88 + 1.61i)T + (1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (3.63 - 2.22i)T + (4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (5.05 + 0.398i)T + (12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (-5.49 - 1.32i)T + (15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (4.35 - 0.342i)T + (18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (-0.132 - 0.408i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.956 - 0.229i)T + (25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (-4.34 + 5.97i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.54 + 1.84i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (8.36 + 4.26i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (2.89 + 3.39i)T + (-7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (0.618 + 2.57i)T + (-47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (4.30 - 1.39i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (6.84 + 13.4i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (11.1 + 6.81i)T + (30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (0.933 + 1.52i)T + (-32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (5.64 - 5.64i)T - 73iT^{2} \) |
| 79 | \( 1 + (-4.05 - 1.68i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 8.64iT - 83T^{2} \) |
| 89 | \( 1 + (-11.7 + 13.7i)T + (-13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (1.38 - 2.25i)T + (-44.0 - 86.4i)T^{2} \) |
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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
−9.986946464349366392480367850759, −9.393056294946020588286120304275, −8.166477433997555807333374862221, −7.42629358680637575446916507894, −6.12231788227078150508137684751, −5.06575561230862834088646731142, −4.50161108126467461208332576277, −3.20244012668964241812177484200, −1.96711518019326376355182504917, −0.33605028120003504096889670571,
2.70070040688516010059156468942, 3.08654780518188777974497331883, 4.72021111033730694818981237998, 5.69522822671108130790801375870, 6.41838242702618239997206439596, 7.29884803650870130512461776628, 8.038009311086714560792099007606, 9.136889772644661128519347295178, 10.09329729197301245251225369328, 10.61369837029663891563473475346
