| L(s) = 1 | + (−1.40 − 0.139i)2-s + (−0.342 + 0.939i)3-s + (1.96 + 0.392i)4-s + (−3.00 + 0.529i)5-s + (0.612 − 1.27i)6-s + (1.26 − 2.18i)7-s + (−2.70 − 0.825i)8-s + (−0.766 − 0.642i)9-s + (4.30 − 0.327i)10-s + (2.96 − 1.71i)11-s + (−1.03 + 1.70i)12-s + (1.65 + 4.54i)13-s + (−2.07 + 2.89i)14-s + (0.529 − 3.00i)15-s + (3.69 + 1.53i)16-s + (2.60 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 − 0.0985i)2-s + (−0.197 + 0.542i)3-s + (0.980 + 0.196i)4-s + (−1.34 + 0.237i)5-s + (0.249 − 0.520i)6-s + (0.476 − 0.825i)7-s + (−0.956 − 0.291i)8-s + (−0.255 − 0.214i)9-s + (1.36 − 0.103i)10-s + (0.893 − 0.515i)11-s + (−0.300 + 0.493i)12-s + (0.458 + 1.26i)13-s + (−0.555 + 0.774i)14-s + (0.136 − 0.776i)15-s + (0.923 + 0.384i)16-s + (0.630 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494988 + 0.395029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494988 + 0.395029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.139i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.378 - 4.34i)T \) |
| good | 5 | \( 1 + (3.00 - 0.529i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.18i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.96 + 1.71i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.65 - 4.54i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.60 + 2.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.901 - 5.11i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (6.27 - 7.48i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.05 - 7.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 11.4iT - 37T^{2} \) |
| 41 | \( 1 + (-7.17 - 2.61i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.23 - 0.571i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.30 - 6.12i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.288 - 0.0508i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.52 - 11.3i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-6.55 - 1.15i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.55 + 1.85i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.430 - 2.43i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.38 + 0.505i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.76 - 1.00i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (8.43 + 4.86i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.09 + 2.58i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.565 - 0.474i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.23390355968802121228300559584, −10.54669046173507783326494107597, −9.336476131359669066456437377448, −8.711841782360665752877306960738, −7.51657076273469285202480788758, −7.14165719901335764678707620751, −5.75251482195084909776001959061, −3.98468403257664526248604196509, −3.58226220005434561643091505557, −1.30805027812967144901490604304,
0.63947585461881276827664324184, 2.29090517068310965646037809391, 3.84258099019263639072790260371, 5.40784305048149861542095081091, 6.43544145625643235726853415613, 7.53667463089750645085253629542, 8.143362279640266966942729087769, 8.759862601370845977710086888263, 9.912061038514446834877229260018, 11.10335930208579643174759318188
