L(s) = 1 | + (0.190 + 0.587i)2-s + (0.809 + 0.587i)3-s + (1.30 − 0.951i)4-s + (−0.190 + 0.587i)6-s + (2.42 − 1.76i)7-s + (1.80 + 1.31i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 3.21i)11-s + 1.61·12-s + (−1.5 − 4.61i)13-s + (1.5 + 1.08i)14-s + (0.572 − 1.76i)16-s + (0.454 − 1.40i)17-s + (−0.5 + 0.363i)18-s + (5.85 + 4.25i)19-s + ⋯ |
L(s) = 1 | + (0.135 + 0.415i)2-s + (0.467 + 0.339i)3-s + (0.654 − 0.475i)4-s + (−0.0779 + 0.239i)6-s + (0.917 − 0.666i)7-s + (0.639 + 0.464i)8-s + (0.103 + 0.317i)9-s + (−0.243 + 0.969i)11-s + 0.467·12-s + (−0.416 − 1.28i)13-s + (0.400 + 0.291i)14-s + (0.143 − 0.440i)16-s + (0.110 − 0.339i)17-s + (−0.117 + 0.0856i)18-s + (1.34 + 0.975i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.51899 + 0.427405i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.51899 + 0.427405i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (0.809 - 3.21i)T \) |
good | 2 | \( 1 + (-0.190 - 0.587i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 1.76i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.454 + 1.40i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.85 - 4.25i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + (1.11 - 0.812i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.663 + 2.04i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.73 + 1.26i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.42 + 2.48i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + (-9.39 - 6.82i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.61 + 8.05i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.78 + 6.37i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.16 - 9.73i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + (2.30 - 7.10i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.118 - 0.0857i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.88 - 11.9i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.89 - 12.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + (5.07 + 15.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.04051452112232716506468101193, −9.852846930950155367700538032323, −8.187813358510082442367893956488, −7.67323800600122576411791823277, −7.10843395929492074551079095833, −5.65218694033819430297319202481, −5.09741614886881604890788705933, −4.00459623156620071522734985081, −2.62979174371708060695697148657, −1.44239997242976584315634446778,
1.58118377648209281599832567684, 2.48890871084529067602339968019, 3.48110919674696418983579898815, 4.65069950605593924050717534739, 5.82909025089776788515344878043, 6.88985406217527221731495307680, 7.68081055776470193974420497957, 8.432574095669745827244429452913, 9.198310638762687598312070219386, 10.29872826355200795197810155911