L(s) = 1 | + (1.43 − 2.48i)5-s + (−0.736 + 2.54i)7-s + (2.34 − 1.35i)11-s + 3.68i·13-s + (3.22 + 5.58i)17-s + (2.73 + 1.58i)19-s + (−2.59 − 1.49i)23-s + (−1.61 − 2.79i)25-s − 2.86i·29-s + (−8.26 + 4.77i)31-s + (5.25 + 5.47i)35-s + (−1.70 + 2.95i)37-s + 1.58·41-s + 9.35·43-s + (−5.65 + 9.79i)47-s + ⋯ |
L(s) = 1 | + (0.641 − 1.11i)5-s + (−0.278 + 0.960i)7-s + (0.708 − 0.408i)11-s + 1.02i·13-s + (0.781 + 1.35i)17-s + (0.628 + 0.362i)19-s + (−0.540 − 0.311i)23-s + (−0.322 − 0.558i)25-s − 0.532i·29-s + (−1.48 + 0.857i)31-s + (0.888 + 0.925i)35-s + (−0.280 + 0.485i)37-s + 0.248·41-s + 1.42·43-s + (−0.824 + 1.42i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.968544843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.968544843\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.736 - 2.54i)T \) |
good | 5 | \( 1 + (-1.43 + 2.48i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.68iT - 13T^{2} \) |
| 17 | \( 1 + (-3.22 - 5.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.73 - 1.58i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.86iT - 29T^{2} \) |
| 31 | \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 - 9.35T + 43T^{2} \) |
| 47 | \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.16 + 1.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 7.51i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.566 - 0.327i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.86 + 6.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.86iT - 71T^{2} \) |
| 73 | \( 1 + (-11.0 + 6.39i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + (-3.14 + 5.45i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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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
−9.134797747581061504521892829691, −8.559082544553587836542551049545, −7.73619677257401404267779359933, −6.49430933041358188675865541120, −5.90465102419764675057124630157, −5.31749679755662834804765307157, −4.28209360628589867556360152354, −3.40130179187864372135960098578, −2.02111303338941674428499980458, −1.28135322934874205881713402077,
0.73809717405247040542469163815, 2.16634884102049849697575294265, 3.20275646378729716985378922426, 3.80571489347433598129728401531, 5.10880361790516529173405205560, 5.83169056927189900497130379984, 6.81634081567850693334302254929, 7.26212978829365249920770300692, 7.88595277154432198276741760526, 9.325478810598909915616335137862