L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (2 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (−2.5 + 4.33i)11-s − 5·13-s + 0.999·15-s + (2 − 3.46i)17-s + (−3.5 − 6.06i)19-s + (−2.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (−1 + 1.73i)31-s + (−2.5 − 4.33i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.753 + 1.30i)11-s − 1.38·13-s + 0.258·15-s + (0.485 − 0.840i)17-s + (−0.802 − 1.39i)19-s + (−0.545 + 0.188i)21-s + (0.104 + 0.180i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + (−0.179 + 0.311i)31-s + (−0.435 − 0.753i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1597855119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1597855119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + (5.5 + 9.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.133420932104137890376915145369, −8.330917458150082397045020997593, −7.42940549279839280970164998894, −6.83208236212602830687955128796, −5.29027389718820248794731076056, −5.03576302052680691499671645911, −4.37820587345781306290234564019, −2.84697869618795500349647197762, −1.98869150420860494146811865789, −0.06082354070551992358364437882,
1.45164570094999445784551396447, 2.65733814442995724574577673850, 3.74279348049847136373640752177, 4.76188174295819758240208273876, 5.68974836382059332668367513598, 6.39754611974417693666583111571, 7.48793522891975515262992772740, 7.921382372792288363445706629482, 8.520793907511696753380657272931, 9.875008337117172300089976161051