L(s) = 1 | + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.326 + 1.85i)5-s + (−0.711 − 4.03i)7-s + (0.500 − 0.866i)8-s + (0.939 + 1.62i)10-s + (1.67 − 2.89i)11-s + (1.48 − 1.24i)13-s + (−2.04 − 3.55i)14-s + (0.173 − 0.984i)16-s + (−3.40 − 2.85i)17-s + (0.0932 + 0.0339i)19-s + (1.43 + 1.20i)20-s + (0.580 − 3.29i)22-s + (4.76 + 8.24i)23-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (0.383 − 0.321i)4-s + (0.145 + 0.827i)5-s + (−0.269 − 1.52i)7-s + (0.176 − 0.306i)8-s + (0.297 + 0.514i)10-s + (0.503 − 0.872i)11-s + (0.411 − 0.345i)13-s + (−0.547 − 0.948i)14-s + (0.0434 − 0.246i)16-s + (−0.826 − 0.693i)17-s + (0.0213 + 0.00778i)19-s + (0.321 + 0.270i)20-s + (0.123 − 0.701i)22-s + (0.992 + 1.71i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95710 - 1.08811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95710 - 1.08811i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (3.50 + 4.97i)T \) |
good | 5 | \( 1 + (-0.326 - 1.85i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.711 + 4.03i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.67 + 2.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 1.24i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.40 + 2.85i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.0932 - 0.0339i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-4.76 - 8.24i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.387 + 0.670i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 41 | \( 1 + (4.36 - 3.66i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + (3.55 + 6.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.806i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (1.40 - 7.97i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (8.19 - 6.87i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.28 - 12.9i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.79 + 1.01i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 + (0.943 + 5.35i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.504 + 0.423i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.612 - 3.47i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.81 + 3.13i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.55059306737152198703615677386, −9.854127480265660345410884126637, −8.682272448768393196992692808958, −7.35013227583370331654224120787, −6.83169675189239457364222632765, −5.96019031951350329626555188925, −4.68282446963819053278327007279, −3.62578949316181445674247134369, −2.94231431208544997787617027697, −1.07510387842674868010768773128,
1.80641380633240933061714379314, 2.96587343376994681652234832587, 4.51300129750981284462623850526, 4.99996471551196810114014435884, 6.28323813848785324754089599806, 6.67228691620217828317489279650, 8.336062415715434325897268237979, 8.756385989398218096258083460052, 9.590548099683586945074457302953, 10.79738289203314036383972370110