L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + 1.00i·9-s + (−0.258 − 0.965i)10-s + (0.965 − 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s + (0.866 − 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s + 1.93·19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + 1.00i·9-s + (−0.258 − 0.965i)10-s + (0.965 − 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.499 + 0.866i)14-s + (0.866 − 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s + 1.93·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.523906948\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.523906948\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.93T + T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−9.200973878681045513473804342199, −8.557927068896963839810548455493, −7.55629787522670288762448998219, −6.32638470841691147050429811942, −5.74880623684087789000133425742, −4.94048191738381606017622126111, −4.08711375186388231047851359786, −3.46095798864133262589731421900, −2.50820806093645604114912952291, −1.45088028572874639964028625583,
1.63259442846936896478816603866, 3.05058794389736640794544758194, 3.24864745378742415622737644037, 4.07901103066287607152791798540, 5.75945234181353388199129728830, 6.00779338616854954332594074719, 6.90061709815980933083759441381, 7.52632917292044583383268679897, 7.963744412762845729888346002972, 9.065383153417079372109922135540