| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.512 + 0.329i)5-s + (0.654 − 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.0867 − 0.603i)10-s + (0.0366 + 0.0801i)11-s + (0.415 + 0.909i)12-s + (−0.291 − 2.02i)13-s + (−0.841 − 0.540i)14-s + (0.584 − 0.171i)15-s + (−0.142 + 0.989i)16-s + (0.448 − 0.517i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.229 + 0.147i)5-s + (0.267 − 0.308i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0274 − 0.190i)10-s + (0.0110 + 0.0241i)11-s + (0.119 + 0.262i)12-s + (−0.0808 − 0.562i)13-s + (−0.224 − 0.144i)14-s + (0.150 − 0.0443i)15-s + (−0.0355 + 0.247i)16-s + (0.108 − 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123331 + 0.549355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123331 + 0.549355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (3.50 + 3.27i)T \) |
| good | 5 | \( 1 + (0.512 - 0.329i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-0.0366 - 0.0801i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.291 + 2.02i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.448 + 0.517i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.55 - 5.26i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.230 + 0.265i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (5.95 - 1.74i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-7.13 - 4.58i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (3.96 - 2.54i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.36 + 0.400i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 + (1.03 - 7.17i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.496 - 3.45i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (11.1 - 3.28i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (2.97 - 6.51i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (5.66 - 12.4i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.0217 - 0.0251i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.68 - 11.7i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (12.0 + 7.74i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.61 + 1.64i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-10.8 + 6.99i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.16452120939082458689473140470, −9.656501834416934451933900175926, −8.522777816799905050481790616480, −7.76629515487875669122406790155, −7.09195190803595172281545363940, −5.97446230141947248612309559012, −5.52796613978898116011316461586, −4.38891638820892912207846641599, −3.13012845360625262660901069750, −1.43954668933624380715257303040,
0.32645905527197898126913246221, 1.81724973309312619233118224303, 3.28406205716277659572634212386, 4.24223738393496693387711577717, 5.09383114937063691910175743716, 6.21986916454323334975567156956, 7.26261751565342466226905419968, 7.975861036597905498780626619240, 9.167905526534105162311328067567, 9.659559554182833074900171074281
