| L(s) = 1 | + (0.248 + 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (−1.80 − 1.32i)5-s + (0.425 + 0.904i)6-s + (1.23 − 1.70i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (0.834 − 2.07i)10-s + (−4.81 + 1.23i)11-s + (−0.770 + 0.637i)12-s + (−1.73 − 4.39i)13-s + (1.96 + 0.776i)14-s + (−2.01 − 0.962i)15-s + (0.535 − 0.844i)16-s + (0.599 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (0.175 + 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.805 − 0.592i)5-s + (0.173 + 0.369i)6-s + (0.468 − 0.644i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (0.263 − 0.656i)10-s + (−1.45 + 0.372i)11-s + (−0.222 + 0.184i)12-s + (−0.482 − 1.21i)13-s + (0.524 + 0.207i)14-s + (−0.521 − 0.248i)15-s + (0.133 − 0.211i)16-s + (0.145 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.873420 - 0.697654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.873420 - 0.697654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.248 - 0.968i)T \) |
| 3 | \( 1 + (-0.982 + 0.187i)T \) |
| 5 | \( 1 + (1.80 + 1.32i)T \) |
| good | 7 | \( 1 + (-1.23 + 1.70i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (4.81 - 1.23i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (1.73 + 4.39i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-0.599 + 1.09i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.960 + 5.03i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (5.68 + 0.357i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-4.79 - 0.605i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (0.369 + 0.203i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (8.73 + 5.54i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.222 + 3.53i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-4.39 + 1.42i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-3.28 + 3.50i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-4.58 - 2.15i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-4.66 - 5.63i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.221 + 3.51i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (1.52 + 12.0i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (10.2 + 9.65i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-8.74 - 7.23i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 12.3i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-3.58 - 0.684i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (9.71 - 11.7i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-0.190 + 1.51i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.17412871180315799787983185210, −9.021917235745020030599436170277, −8.151689747691609725191985842800, −7.64555831682963180500887973378, −7.10384439241736157438835238532, −5.43401428242453023271255266436, −4.83917127089357280451969738741, −3.82530727697116677120332079057, −2.59900528633274206696992725920, −0.48708119615149388126054343706,
1.98337174811586570339998332918, 2.89847437391242287840768700295, 3.91990050593361245604793719947, 4.86979090369073989775094035644, 5.98620938727296549100235820851, 7.31250205081534775276008176701, 8.167467906196046691616724955445, 8.675071682978311529464727536270, 10.02730919625787660564814543346, 10.38348491984042904426761697895
