| L(s) = 1 | + (1.40 − 0.130i)2-s + (−1.50 − 0.851i)3-s + (1.96 − 0.367i)4-s + (1.73 − 3.52i)5-s + (−2.23 − 1.00i)6-s + (−1.31 − 1.71i)7-s + (2.72 − 0.774i)8-s + (1.55 + 2.56i)9-s + (1.98 − 5.19i)10-s + (−1.86 + 2.12i)11-s + (−3.27 − 1.11i)12-s + (0.0127 − 0.195i)13-s + (−2.07 − 2.24i)14-s + (−5.62 + 3.84i)15-s + (3.72 − 1.44i)16-s + (−1.83 − 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.995 − 0.0922i)2-s + (−0.870 − 0.491i)3-s + (0.982 − 0.183i)4-s + (0.777 − 1.57i)5-s + (−0.912 − 0.408i)6-s + (−0.497 − 0.648i)7-s + (0.961 − 0.273i)8-s + (0.517 + 0.855i)9-s + (0.628 − 1.64i)10-s + (−0.562 + 0.641i)11-s + (−0.946 − 0.322i)12-s + (0.00354 − 0.0541i)13-s + (−0.555 − 0.600i)14-s + (−1.45 + 0.991i)15-s + (0.932 − 0.361i)16-s + (−0.446 − 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29336 - 1.73074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29336 - 1.73074i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.130i)T \) |
| 3 | \( 1 + (1.50 + 0.851i)T \) |
| good | 5 | \( 1 + (-1.73 + 3.52i)T + (-3.04 - 3.96i)T^{2} \) |
| 7 | \( 1 + (1.31 + 1.71i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (1.86 - 2.12i)T + (-1.43 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0127 + 0.195i)T + (-12.8 - 1.69i)T^{2} \) |
| 17 | \( 1 + (1.83 + 1.83i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.326 + 0.218i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (3.00 - 3.91i)T + (-5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-8.17 - 2.77i)T + (23.0 + 17.6i)T^{2} \) |
| 31 | \( 1 + (-3.06 - 5.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.93 + 2.90i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.44 + 1.10i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (-7.00 + 7.99i)T + (-5.61 - 42.6i)T^{2} \) |
| 47 | \( 1 + (2.12 + 7.94i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.19 - 11.0i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-7.88 - 3.88i)T + (35.9 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-10.2 - 3.46i)T + (48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-2.11 - 2.40i)T + (-8.74 + 66.4i)T^{2} \) |
| 71 | \( 1 + (7.34 + 3.04i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.73 - 1.13i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.67 - 13.6i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (1.12 + 2.27i)T + (-50.5 + 65.8i)T^{2} \) |
| 89 | \( 1 + (7.68 + 3.18i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-14.0 - 8.13i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.39382370418480863114981992799, −10.03968015839694001310801780127, −8.658079349529077219110839351853, −7.41592145843671940259266592745, −6.64195089435522399442756385979, −5.57349142599807131653344484887, −5.02837410049331332577277733936, −4.16187470810847333189593292545, −2.24778556940490738645252825348, −1.02203154075893294307074256838,
2.39824399703991149812536053029, 3.21475942186213023897826616459, 4.48373148979160013472378360843, 5.74722185116099248211823002195, 6.25711417832551202641036773142, 6.72790352057912363303632484043, 8.130701349025139233074162624927, 9.746501520681905279125363656975, 10.33614526648705015320965073487, 11.04894090306891891289282590727
