L(s) = 1 | + (−30.5 − 30.5i)2-s + (−54.5 − 129. i)3-s + 1.35e3i·4-s + (1.26e3 + 602. i)5-s + (−2.28e3 + 5.60e3i)6-s + (−2.74e3 + 2.74e3i)7-s + (2.56e4 − 2.56e4i)8-s + (−1.37e4 + 1.40e4i)9-s + (−2.01e4 − 5.68e4i)10-s + 4.81e3i·11-s + (1.74e5 − 7.36e4i)12-s + (6.67e4 + 6.67e4i)13-s + 1.67e5·14-s + (9.12e3 − 1.95e5i)15-s − 8.71e5·16-s + (−3.46e4 − 3.46e4i)17-s + ⋯ |
L(s) = 1 | + (−1.34 − 1.34i)2-s + (−0.388 − 0.921i)3-s + 2.63i·4-s + (0.902 + 0.430i)5-s + (−0.718 + 1.76i)6-s + (−0.432 + 0.432i)7-s + (2.21 − 2.21i)8-s + (−0.698 + 0.715i)9-s + (−0.635 − 1.79i)10-s + 0.0991i·11-s + (2.43 − 1.02i)12-s + (0.648 + 0.648i)13-s + 1.16·14-s + (0.0465 − 0.998i)15-s − 3.32·16-s + (−0.100 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.660484 - 0.204351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660484 - 0.204351i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (54.5 + 129. i)T \) |
| 5 | \( 1 + (-1.26e3 - 602. i)T \) |
good | 2 | \( 1 + (30.5 + 30.5i)T + 512iT^{2} \) |
| 7 | \( 1 + (2.74e3 - 2.74e3i)T - 4.03e7iT^{2} \) |
| 11 | \( 1 - 4.81e3iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-6.67e4 - 6.67e4i)T + 1.06e10iT^{2} \) |
| 17 | \( 1 + (3.46e4 + 3.46e4i)T + 1.18e11iT^{2} \) |
| 19 | \( 1 - 4.27e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-6.70e5 + 6.70e5i)T - 1.80e12iT^{2} \) |
| 29 | \( 1 - 8.83e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (5.43e6 - 5.43e6i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 - 2.76e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (1.38e7 + 1.38e7i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.97e6 - 1.97e6i)T + 1.11e15iT^{2} \) |
| 53 | \( 1 + (1.15e7 - 1.15e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 - 2.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.28e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (9.09e7 - 9.09e7i)T - 2.72e16iT^{2} \) |
| 71 | \( 1 + 4.40e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-3.10e8 - 3.10e8i)T + 5.88e16iT^{2} \) |
| 79 | \( 1 - 3.03e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (-4.46e8 + 4.46e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 1.06e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + (7.40e8 - 7.40e8i)T - 7.60e17iT^{2} \) |
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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
−17.60683268894353380547276512535, −16.57003231670163543272927242259, −13.60529399213196653121050474722, −12.42877507945084369702240260050, −11.22487352635968036838352102709, −9.917587916774334408280452768832, −8.475594512208862974520226932237, −6.63654124582845958602623574311, −2.68875623403361099318464737949, −1.33880598422250727983123895349,
0.66738072095768990348163283088, 5.26509243513918076873623245517, 6.51558948283215179864610366058, 8.626626003093648444428330815472, 9.723380383751360618901059090869, 10.69881690862611281820998266221, 13.73688006290114518731160806644, 15.28137708463497524795781559428, 16.21513143126234924483875014714, 17.21381307093299755559232677479