L(s) = 1 | + (1.96 − 1.13i)3-s + (−1.5 + 0.866i)5-s + 2i·7-s + (1.08 − 1.87i)9-s + (−2.85 + 4.94i)13-s + (−1.96 + 3.40i)15-s + (−0.583 − 1.01i)17-s + (−0.144 + 4.35i)19-s + (2.27 + 3.93i)21-s + (4.31 + 2.49i)23-s + (−1 + 1.73i)25-s + 1.89i·27-s + (0.583 − 1.01i)29-s + 0.289·31-s + (−1.73 − 3i)35-s + ⋯ |
L(s) = 1 | + (1.13 − 0.656i)3-s + (−0.670 + 0.387i)5-s + 0.755i·7-s + (0.361 − 0.625i)9-s + (−0.792 + 1.37i)13-s + (−0.508 + 0.880i)15-s + (−0.141 − 0.245i)17-s + (−0.0331 + 0.999i)19-s + (0.496 + 0.859i)21-s + (0.900 + 0.519i)23-s + (−0.200 + 0.346i)25-s + 0.364i·27-s + (0.108 − 0.187i)29-s + 0.0519·31-s + (−0.292 − 0.507i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.765135409\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765135409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.144 - 4.35i)T \) |
good | 3 | \( 1 + (-1.96 + 1.13i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (2.85 - 4.94i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.583 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.31 - 2.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.583 + 1.01i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.289T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + (-8.56 + 4.94i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.380 - 0.659i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.90 + 3.40i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.333 - 0.577i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.51 - 5.49i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.31 + 3.07i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.18 + 5.30i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.57 - 9.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 + 6.65i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.289T + 83T^{2} \) |
| 89 | \( 1 + (-14.0 - 8.12i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.56 + 4.94i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.426682527025157549968922958537, −9.128997938062656730061201120387, −8.156185965898588682591187911819, −7.48591865401578288192888748361, −6.92397949008033622062847192441, −5.79976282542872610876043436922, −4.57081840833987445878268856534, −3.51361872837547419688705549601, −2.60834668899821059259680579314, −1.73506144375670255960925285181,
0.64626849554689006492719609957, 2.61498894599477890969755766954, 3.32638084137940066648654809510, 4.36705253122293527917516290815, 4.88155579296374408059589427162, 6.32239072981061449561151120694, 7.57219873334937429897454936188, 7.888975053469769335537593837222, 8.820655927547581768742177498588, 9.440753368343015129742857547218