| L(s) = 1 | + (−0.925 + 1.06i)2-s + (−0.318 − 0.217i)3-s + (−0.285 − 1.97i)4-s + (−2.23 − 0.167i)5-s + (0.527 − 0.139i)6-s + (2.62 − 0.318i)7-s + (2.38 + 1.52i)8-s + (−1.04 − 2.65i)9-s + (2.24 − 2.23i)10-s + (4.62 + 1.81i)11-s + (−0.339 + 0.692i)12-s + (4.05 − 3.23i)13-s + (−2.09 + 3.10i)14-s + (0.675 + 0.538i)15-s + (−3.83 + 1.13i)16-s + (−1.83 − 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.184 − 0.125i)3-s + (−0.142 − 0.989i)4-s + (−0.998 − 0.0748i)5-s + (0.215 − 0.0569i)6-s + (0.992 − 0.120i)7-s + (0.841 + 0.540i)8-s + (−0.347 − 0.884i)9-s + (0.710 − 0.705i)10-s + (1.39 + 0.547i)11-s + (−0.0979 + 0.200i)12-s + (1.12 − 0.897i)13-s + (−0.558 + 0.829i)14-s + (0.174 + 0.139i)15-s + (−0.959 + 0.282i)16-s + (−0.445 − 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.783609 - 0.0334839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.783609 - 0.0334839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.925 - 1.06i)T \) |
| 7 | \( 1 + (-2.62 + 0.318i)T \) |
| good | 3 | \( 1 + (0.318 + 0.217i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (2.23 + 0.167i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-4.62 - 1.81i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.05 + 3.23i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.83 + 1.97i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.20 + 2.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.13 + 2.29i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.93 - 8.48i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.0585 - 0.101i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.160 + 0.0494i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (4.98 + 10.3i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.97 + 4.09i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (10.2 - 1.54i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-2.75 + 0.850i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.758 - 10.1i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.62 - 8.51i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (5.07 - 2.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.49 - 2.16i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 7.37i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-11.9 - 6.92i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.95 + 2.45i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (1.74 - 0.684i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 1.56iT - 97T^{2} \) |
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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
−12.18580905845330917126342474402, −11.43128391697142906483020516600, −10.61352311619570416411641059421, −9.040082233303129324680352847195, −8.560739304282329393554353659131, −7.35600127166260757109555536761, −6.55236938268697636778500257895, −5.16043563273760311897836698210, −3.86890851423148176214128344192, −1.05974346359626297485786138455,
1.62341411520913351249910895030, 3.63359764209158424739184669782, 4.54203956821445050933237764688, 6.44255305665694440388119055116, 7.989433049087123970633602604150, 8.362165584010205082204063422297, 9.522327287036003405498955572962, 11.04051669843623766861985653371, 11.38332573142840841394170798115, 11.88173138296845707128044000020
