| L(s) = 1 | + (−1.97 − 0.718i)3-s + (−1.01 − 5.73i)5-s + (2.37 − 4.10i)7-s + (−17.3 − 14.5i)9-s + (−32.1 − 55.6i)11-s + (24.9 − 9.06i)13-s + (−2.12 + 12.0i)15-s + (42.3 − 35.5i)17-s + (−81.6 + 14.0i)19-s + (−7.62 + 6.40i)21-s + (−10.9 + 62.3i)23-s + (85.6 − 31.1i)25-s + (52.0 + 90.1i)27-s + (58.3 + 48.9i)29-s + (−43.7 + 75.8i)31-s + ⋯ |
| L(s) = 1 | + (−0.379 − 0.138i)3-s + (−0.0904 − 0.512i)5-s + (0.128 − 0.221i)7-s + (−0.640 − 0.537i)9-s + (−0.880 − 1.52i)11-s + (0.531 − 0.193i)13-s + (−0.0365 + 0.207i)15-s + (0.603 − 0.506i)17-s + (−0.985 + 0.169i)19-s + (−0.0792 + 0.0665i)21-s + (−0.0996 + 0.565i)23-s + (0.684 − 0.249i)25-s + (0.371 + 0.642i)27-s + (0.373 + 0.313i)29-s + (−0.253 + 0.439i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.540782 - 0.788361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.540782 - 0.788361i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (81.6 - 14.0i)T \) |
| good | 3 | \( 1 + (1.97 + 0.718i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (1.01 + 5.73i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-2.37 + 4.10i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (32.1 + 55.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-24.9 + 9.06i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-42.3 + 35.5i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (10.9 - 62.3i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-58.3 - 48.9i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (43.7 - 75.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 58.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-128. - 46.8i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 65.2i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (154. + 129. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-125. + 711. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-282. + 237. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-30.1 + 170. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (621. + 521. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (102. + 583. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (288. + 105. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-724. - 263. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (120. - 208. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-909. + 331. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (925. - 776. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−13.59442768018417060185455666633, −12.59337582484349183262441774440, −11.43285682001550920654809787243, −10.57384636029212175116665316151, −8.916150739929509281353565696437, −8.043396167668620467845829964573, −6.29052128136262005443996580925, −5.21971921194795396334814545880, −3.29286178403988412843113349558, −0.63441427295872015669210745440,
2.43932668122773913666462817210, 4.52459631434858613794238987459, 5.90040300002437677395314751046, 7.33621643819931774158414962889, 8.568851172449537225347509347239, 10.18903857625826213148100704988, 10.89829547922726876421955348953, 12.13835094352815135189945308881, 13.17058738168115198602217818986, 14.55093473184517418694479894559
