L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 − 0.707i)7-s + 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s − 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s − 0.999·20-s + 0.517·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 − 0.707i)7-s + 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s − 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s − 0.999·20-s + 0.517·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1541896512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1541896512\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 1.93iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 0.517T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.417999007765749515096617406453, −7.983046594767630421045149449992, −7.57362403004777823815086223848, −6.38178118112275484337031252679, −5.78103023624020710263935122403, −4.95690558064563698008449152782, −3.86917804985841868341860292505, −2.85492217810657940321506184939, −1.32282367428908738240996956367, −0.13582808960997706155109265551,
2.18427489676889805561817834551, 2.47196114522602099103383422768, 3.79471641459794778772694857529, 4.31766302391634577695401882519, 5.97137640106841805237740299726, 6.67431649265759575141372654150, 7.15669526572573724566832673945, 8.114120179590897501564808655457, 8.832964883092729035861564188597, 9.447748013620951530024054533180