L(s) = 1 | + (−1.22 + 1.22i)2-s + 1.00i·4-s + (−4.89 + 4.89i)7-s + (−6.12 − 6.12i)8-s + 3·11-s + (−7.34 − 7.34i)13-s − 11.9i·14-s + 10.9·16-s + (−13.4 + 13.4i)17-s − 5i·19-s + (−3.67 + 3.67i)22-s + (−17.1 − 17.1i)23-s + 18·26-s + (−4.89 − 4.89i)28-s − 30i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.612i)2-s + 0.250i·4-s + (−0.699 + 0.699i)7-s + (−0.765 − 0.765i)8-s + 0.272·11-s + (−0.565 − 0.565i)13-s − 0.857i·14-s + 0.687·16-s + (−0.792 + 0.792i)17-s − 0.263i·19-s + (−0.167 + 0.167i)22-s + (−0.745 − 0.745i)23-s + 0.692·26-s + (−0.174 − 0.174i)28-s − 1.03i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0597231 - 0.121069i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0597231 - 0.121069i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 1.22i)T - 4iT^{2} \) |
| 7 | \( 1 + (4.89 - 4.89i)T - 49iT^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 + (7.34 + 7.34i)T + 169iT^{2} \) |
| 17 | \( 1 + (13.4 - 13.4i)T - 289iT^{2} \) |
| 19 | \( 1 + 5iT - 361T^{2} \) |
| 23 | \( 1 + (17.1 + 17.1i)T + 529iT^{2} \) |
| 29 | \( 1 + 30iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 + (-19.5 + 19.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 57T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-4.89 - 4.89i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (7.34 - 7.34i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-31.8 - 31.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 90iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28T + 3.72e3T^{2} \) |
| 67 | \( 1 + (47.7 - 47.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 42T + 5.04e3T^{2} \) |
| 73 | \( 1 + (13.4 + 13.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 80iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-111. - 111. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 15iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (53.8 - 53.8i)T - 9.40e3iT^{2} \) |
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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
−12.58546008069906418545767082114, −11.86956891383589005223024314565, −10.46800180106366545235009215937, −9.436142571758504169925928690631, −8.710995006050827522679517770718, −7.74443982720865783610740897097, −6.65053411589934279969974893977, −5.80268265136281967277961176080, −4.03637687945360938964497067194, −2.61123488384729660036919720633,
0.084607343842383066068466514975, 1.84883495312584071143165209526, 3.43481396056161488113640760637, 4.97818828669437133914202143160, 6.36828837680378856227475322096, 7.35171562604295048599959053234, 8.816064839565592963096325085293, 9.593796146917377975242974361027, 10.29639715551975428188743392400, 11.29506231130090077524540759744