L(s) = 1 | + (0.5 − 0.866i)3-s − 4·5-s + (1 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + (−2 + 3.46i)15-s + (−1 − 1.73i)17-s + (1 + 1.73i)19-s + 1.99·21-s + 11·25-s − 0.999·27-s + (3 − 5.19i)29-s − 10·31-s + (−1.99 − 3.46i)33-s + (−4 − 6.92i)35-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s − 1.78·5-s + (0.377 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.516 + 0.894i)15-s + (−0.242 − 0.420i)17-s + (0.229 + 0.397i)19-s + 0.436·21-s + 2.20·25-s − 0.192·27-s + (0.557 − 0.964i)29-s − 1.79·31-s + (−0.348 − 0.603i)33-s + (−0.676 − 1.17i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 4T + 5T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4 - 6.92i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8 + 13.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.539825990130208982825812864153, −8.020775433687687247272169423144, −7.27112693309616845244501986591, −6.50205386759873628864082251007, −5.48163926250373213158095886576, −4.46309268594975672450750702001, −3.57600215762941453280140440722, −2.90517411398229217648747621974, −1.38837792382931733151992885352, 0,
1.66193045617956345981245735836, 3.23790575075785422133145951312, 3.94470945639180785966003757093, 4.46014437005376529873956901530, 5.29642770170814678532703623883, 6.98904904998467218003204509059, 7.15740809612279037781928137880, 8.062270280186357456188813329597, 8.726624634601880262540632241067