| L(s) = 1 | + (22.9 − 6.15i)2-s + (34.8 − 60.4i)3-s + (268. − 155. i)4-s + (−80.1 − 80.1i)5-s + (429. − 1.60e3i)6-s + (249. + 66.9i)7-s + (914. − 914. i)8-s + (847. + 1.46e3i)9-s + (−2.33e3 − 1.34e3i)10-s + (−193. − 723. i)11-s − 2.16e4i·12-s + (2.85e4 − 714. i)13-s + 6.15e3·14-s + (−7.64e3 + 2.04e3i)15-s + (−2.43e4 + 4.21e4i)16-s + (−9.85e4 + 5.68e4i)17-s + ⋯ |
| L(s) = 1 | + (1.43 − 0.384i)2-s + (0.430 − 0.745i)3-s + (1.04 − 0.606i)4-s + (−0.128 − 0.128i)5-s + (0.331 − 1.23i)6-s + (0.104 + 0.0278i)7-s + (0.223 − 0.223i)8-s + (0.129 + 0.223i)9-s + (−0.233 − 0.134i)10-s + (−0.0132 − 0.0494i)11-s − 1.04i·12-s + (0.999 − 0.0250i)13-s + 0.160·14-s + (−0.150 + 0.0404i)15-s + (−0.371 + 0.643i)16-s + (−1.17 + 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.97777 - 1.61981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.97777 - 1.61981i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.85e4 + 714. i)T \) |
| good | 2 | \( 1 + (-22.9 + 6.15i)T + (221. - 128i)T^{2} \) |
| 3 | \( 1 + (-34.8 + 60.4i)T + (-3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (80.1 + 80.1i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + (-249. - 66.9i)T + (4.99e6 + 2.88e6i)T^{2} \) |
| 11 | \( 1 + (193. + 723. i)T + (-1.85e8 + 1.07e8i)T^{2} \) |
| 17 | \( 1 + (9.85e4 - 5.68e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (3.24e4 - 1.20e5i)T + (-1.47e10 - 8.49e9i)T^{2} \) |
| 23 | \( 1 + (9.53e4 + 5.50e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.12e5 + 5.40e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.26e5 + 8.26e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (3.91e5 + 1.46e6i)T + (-3.04e12 + 1.75e12i)T^{2} \) |
| 41 | \( 1 + (-3.80e6 + 1.01e6i)T + (6.91e12 - 3.99e12i)T^{2} \) |
| 43 | \( 1 + (7.43e5 - 4.28e5i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.80e6 - 2.80e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + 6.89e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + (1.47e7 + 3.95e6i)T + (1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (-9.16e6 - 1.58e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-2.61e7 + 7.00e6i)T + (3.51e14 - 2.03e14i)T^{2} \) |
| 71 | \( 1 + (4.29e6 - 1.60e7i)T + (-5.59e14 - 3.22e14i)T^{2} \) |
| 73 | \( 1 + (5.10e6 - 5.10e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 4.76e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.73e7 + 3.73e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + (-2.46e7 - 9.19e7i)T + (-3.40e15 + 1.96e15i)T^{2} \) |
| 97 | \( 1 + (1.64e7 - 6.13e7i)T + (-6.78e15 - 3.91e15i)T^{2} \) |
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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
−18.02843244185912283191788494697, −15.93381946628858405062549945614, −14.45357733728390656024570403680, −13.38672977495422250509738700542, −12.51953101648041361191690428876, −10.97323985153569852723121763352, −8.286308216801865721910053386680, −6.19289782043489197085752254105, −4.13068290293982169861998912448, −2.08993995189452524922352720506,
3.36106117753936992247853090939, 4.75351455031046809296308330228, 6.70713547184313075395834328020, 9.123705143336383438960383913963, 11.19276044113850278806191090391, 12.92958969164547832985167734872, 14.13464143559761492604428635364, 15.32635038991825058740574245834, 16.00351648584383716635198481299, 18.10455729832114395922861012307
