L(s) = 1 | − 1.93·5-s + (−0.866 − 0.5i)13-s + (0.448 − 0.258i)17-s + 2.73·25-s + (−0.448 − 0.258i)29-s + (1.5 + 0.866i)37-s + (−0.965 + 1.67i)41-s + (−0.5 + 0.866i)49-s + 1.93i·53-s + (0.5 + 0.866i)61-s + (1.67 + 0.965i)65-s + i·73-s + (−0.866 + 0.499i)85-s + (0.707 − 1.22i)89-s + (−1.73 + i)97-s + ⋯ |
L(s) = 1 | − 1.93·5-s + (−0.866 − 0.5i)13-s + (0.448 − 0.258i)17-s + 2.73·25-s + (−0.448 − 0.258i)29-s + (1.5 + 0.866i)37-s + (−0.965 + 1.67i)41-s + (−0.5 + 0.866i)49-s + 1.93i·53-s + (0.5 + 0.866i)61-s + (1.67 + 0.965i)65-s + i·73-s + (−0.866 + 0.499i)85-s + (0.707 − 1.22i)89-s + (−1.73 + i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6134785262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6134785262\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + 1.93T + T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.93iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.593441970219817798481818993730, −7.923077507039192574155985896507, −7.56497826406676865796067416347, −6.85338973484458485218776180263, −5.84147077869605373032378110097, −4.74284429635340657471318555727, −4.36872456174880902339675259346, −3.32482396118400615315016909200, −2.75638218972090725794355030397, −0.999996956377009450436392673061,
0.44435091044703936769839552905, 2.09008907314902360738791821798, 3.32289677600560168776593942113, 3.85412034089259137575129601498, 4.65856936448082358543215606803, 5.36359715763413254867407919794, 6.61358300065044925919553261770, 7.22170045328173722531520987867, 7.79549252889581947039103236279, 8.398437758996965241519379621670