L(s) = 1 | + (−1 + 1.73i)5-s + (1.5 − 2.59i)9-s − 6·13-s + (1 + 1.73i)17-s + (0.500 + 0.866i)25-s − 10·29-s + (1 − 1.73i)37-s − 10·41-s + (3 + 5.19i)45-s + (−7 − 12.1i)53-s + (−5 + 8.66i)61-s + (6 − 10.3i)65-s + (−3 − 5.19i)73-s + (−4.5 − 7.79i)81-s − 3.99·85-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.5 − 0.866i)9-s − 1.66·13-s + (0.242 + 0.420i)17-s + (0.100 + 0.173i)25-s − 1.85·29-s + (0.164 − 0.284i)37-s − 1.56·41-s + (0.447 + 0.774i)45-s + (−0.961 − 1.66i)53-s + (−0.640 + 1.10i)61-s + (0.744 − 1.28i)65-s + (−0.351 − 0.608i)73-s + (−0.5 − 0.866i)81-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7 + 12.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−9.272630001372056561032978063335, −8.134059757578399562910137575289, −7.24280425038703764833162872989, −6.93911704071327362505452207204, −5.84164005368389913202312428656, −4.84973105168402477054775524792, −3.81520313091312236138128343933, −3.07867506931942706903209278018, −1.82646294188698968277200242313, 0,
1.64706098684943114573882694571, 2.76504854172385281223470491791, 4.09040925339814885211667780987, 4.88773218873112733949822224837, 5.38475600842983691014409611417, 6.75816902343434762030636003101, 7.59931620616839247962763422443, 7.993176821869416843750214678372, 9.094531240098729751772103102679