L(s) = 1 | + (0.316 − 1.37i)2-s + (−0.556 + 1.64i)3-s + (−1.79 − 0.871i)4-s + (0.0665 − 0.248i)5-s + (2.08 + 1.28i)6-s + (−0.154 + 2.64i)7-s + (−1.77 + 2.20i)8-s + (−2.38 − 1.82i)9-s + (−0.321 − 0.170i)10-s + (1.52 + 5.69i)11-s + (2.43 − 2.46i)12-s + (−1.65 − 1.65i)13-s + (3.59 + 1.04i)14-s + (0.370 + 0.247i)15-s + (2.47 + 3.13i)16-s + (−3.31 + 5.73i)17-s + ⋯ |
L(s) = 1 | + (0.223 − 0.974i)2-s + (−0.321 + 0.946i)3-s + (−0.899 − 0.435i)4-s + (0.0297 − 0.111i)5-s + (0.851 + 0.524i)6-s + (−0.0584 + 0.998i)7-s + (−0.626 + 0.779i)8-s + (−0.793 − 0.608i)9-s + (−0.101 − 0.0538i)10-s + (0.460 + 1.71i)11-s + (0.702 − 0.712i)12-s + (−0.458 − 0.458i)13-s + (0.959 + 0.280i)14-s + (0.0955 + 0.0638i)15-s + (0.619 + 0.784i)16-s + (−0.803 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.821169 + 0.483493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.821169 + 0.483493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.316 + 1.37i)T \) |
| 3 | \( 1 + (0.556 - 1.64i)T \) |
| 7 | \( 1 + (0.154 - 2.64i)T \) |
good | 5 | \( 1 + (-0.0665 + 0.248i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.52 - 5.69i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (1.65 + 1.65i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.31 - 5.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 4.20i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.18 - 3.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.853 - 0.853i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.82 + 2.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.05 - 3.93i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.05iT - 41T^{2} \) |
| 43 | \( 1 + (2.94 + 2.94i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.356 - 0.618i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.18 + 11.8i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.26 - 0.338i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 14.1i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 6.87i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + (-2.87 + 4.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.19 - 3.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.70 - 6.70i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.982 - 0.567i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−11.60609548060884130830642305567, −10.88761743019299729647957247368, −9.811971205366460169717000822766, −9.350389759481994810314210849422, −8.447939182583801987052924323453, −6.63721002311002480257309477060, −5.28311625695617335652485544633, −4.70923442271346606820476504276, −3.45674965737768550133445770452, −2.08065904688438667351491589481,
0.66116627632496816092463588049, 3.12319578233340768978752765051, 4.56357134778397696760956029742, 5.77482081889676044453549035666, 6.70504893823582869185144850190, 7.27813339744569490237362936402, 8.344141286568768147792721063568, 9.160404155762265501074648224124, 10.62212283904106541041070685735, 11.52457513338494324682453582903