| L(s) = 1 | + (1.36 + 0.366i)2-s + (−1.64 + 0.951i)3-s + (1.73 + i)4-s + (1.64 − 4.72i)5-s + (−2.60 + 0.696i)6-s + (11.1 − 2.98i)7-s + (1.99 + 2i)8-s + (−2.68 + 4.65i)9-s + (3.97 − 5.84i)10-s + (4.34 + 1.16i)11-s − 3.80·12-s + (3.12 + 12.6i)13-s + 16.3·14-s + (1.78 + 9.35i)15-s + (1.99 + 3.46i)16-s + (8.80 − 15.2i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.549 + 0.317i)3-s + (0.433 + 0.250i)4-s + (0.328 − 0.944i)5-s + (−0.433 + 0.116i)6-s + (1.59 − 0.427i)7-s + (0.249 + 0.250i)8-s + (−0.298 + 0.517i)9-s + (0.397 − 0.584i)10-s + (0.395 + 0.105i)11-s − 0.317·12-s + (0.240 + 0.970i)13-s + 1.16·14-s + (0.118 + 0.623i)15-s + (0.124 + 0.216i)16-s + (0.518 − 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.02119 + 0.211287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02119 + 0.211287i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 5 | \( 1 + (-1.64 + 4.72i)T \) |
| 13 | \( 1 + (-3.12 - 12.6i)T \) |
| good | 3 | \( 1 + (1.64 - 0.951i)T + (4.5 - 7.79i)T^{2} \) |
| 7 | \( 1 + (-11.1 + 2.98i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-4.34 - 1.16i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-8.80 + 15.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (17.6 - 4.73i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (7.54 + 13.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (3.26 + 5.65i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (20.0 + 20.0i)T + 961iT^{2} \) |
| 37 | \( 1 + (12.7 - 47.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (15.3 - 57.3i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (8.81 - 15.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (30.5 + 30.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 49.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (9.71 + 36.2i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (22.2 - 38.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-32.1 - 8.62i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (61.4 - 16.4i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (67.5 + 67.5i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 139.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-80.8 + 80.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-152. - 40.9i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (1.66 + 6.20i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−13.27481150904931165881214240419, −11.82768461396583940966650619611, −11.45670608878138901024150906935, −10.22025435552083645401344819997, −8.700634482070358037747401928322, −7.74619726815485934787456973796, −6.13949510217359293461266309853, −4.89177513730139606344250174039, −4.43059603104870429283507447129, −1.76221013004627018460180098317,
1.81617462570389623697164111176, 3.58452190724883834924470238324, 5.36277714569037194911716335624, 6.10019176660507098766982200002, 7.39856027997088782724600896867, 8.702259897761502931021493788280, 10.47940375457546227085994790644, 11.12087186279663650910458856221, 11.95133984178650931480979230595, 12.90601302016714919710139711249
