| L(s) = 1 | + (0.930 + 0.365i)2-s + (−1.29 + 1.14i)3-s + (0.733 + 0.680i)4-s + (1.59 + 1.08i)5-s + (−1.62 + 0.592i)6-s + (1.91 − 1.82i)7-s + (0.433 + 0.900i)8-s + (0.371 − 2.97i)9-s + (1.08 + 1.59i)10-s + (0.652 + 4.32i)11-s + (−1.73 − 0.0426i)12-s + (−2.72 + 2.17i)13-s + (2.44 − 1.00i)14-s + (−3.31 + 0.416i)15-s + (0.0747 + 0.997i)16-s + (2.73 + 0.844i)17-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.258i)2-s + (−0.749 + 0.661i)3-s + (0.366 + 0.340i)4-s + (0.712 + 0.485i)5-s + (−0.664 + 0.242i)6-s + (0.722 − 0.691i)7-s + (0.153 + 0.318i)8-s + (0.123 − 0.992i)9-s + (0.343 + 0.503i)10-s + (0.196 + 1.30i)11-s + (−0.499 − 0.0123i)12-s + (−0.754 + 0.601i)13-s + (0.654 − 0.268i)14-s + (−0.855 + 0.107i)15-s + (0.0186 + 0.249i)16-s + (0.664 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34468 + 1.05543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34468 + 1.05543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.930 - 0.365i)T \) |
| 3 | \( 1 + (1.29 - 1.14i)T \) |
| 7 | \( 1 + (-1.91 + 1.82i)T \) |
| good | 5 | \( 1 + (-1.59 - 1.08i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.652 - 4.32i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.72 - 2.17i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 0.844i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (4.83 - 2.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.92 + 6.24i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-6.29 + 1.43i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.87 + 1.66i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.08 + 6.57i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-5.79 + 2.79i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.565 + 0.272i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (0.490 - 1.24i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (3.51 - 3.78i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (1.59 - 1.08i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (9.97 + 10.7i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.49 + 4.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.32 + 1.89i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-10.7 + 4.20i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-0.441 - 0.764i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.56 - 8.23i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (1.87 + 0.282i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−12.17335864272516049765579648063, −10.91018584395858224301595421693, −10.30984592304910623505845394682, −9.486271326474274214005131443534, −7.86456000381261952531622647343, −6.76722248866957524251252735648, −6.01777601458709579702542469490, −4.67308723613182452350255786337, −4.17409098586112036432939774405, −2.16446244717141067116795506188,
1.34468321173261049820832692205, 2.75396073599505285247385324858, 4.76905257055712553344686389694, 5.57276138316687901034628800949, 6.18384715745135676317642109283, 7.61790922423992083881642963124, 8.643577197422926588699723847029, 9.907824369793490924325433905452, 11.03790356042117568903662515340, 11.65515670811672361864029333291
