L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 3.86·5-s − 0.999·8-s + (1.93 + 3.34i)10-s + 3.73·11-s + (3.34 + 5.79i)13-s + (−0.5 − 0.866i)16-s + (−2.70 − 4.69i)17-s + (−1.48 + 2.56i)19-s + (−1.93 + 3.34i)20-s + (1.86 + 3.23i)22-s + 1.46·23-s + 9.92·25-s + (−3.34 + 5.79i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.72·5-s − 0.353·8-s + (0.610 + 1.05i)10-s + 1.12·11-s + (0.928 + 1.60i)13-s + (−0.125 − 0.216i)16-s + (−0.656 − 1.13i)17-s + (−0.340 + 0.589i)19-s + (−0.431 + 0.748i)20-s + (0.397 + 0.689i)22-s + 0.305·23-s + 1.98·25-s + (−0.656 + 1.13i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.269978430\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.269978430\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.86T + 5T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + (-3.34 - 5.79i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.70 + 4.69i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.896 + 1.55i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.637 - 1.10i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.86 - 3.23i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.27 + 9.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.46 + 2.53i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.31 + 7.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.48 - 6.03i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.76 - 4.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + (3.41 + 5.91i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.46 - 4.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.95 + 15.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.94 - 5.10i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.111377788240801438210756390898, −8.483801131106105438582019204547, −7.07602615961765014807879600621, −6.57365465991284065808329274852, −6.14554483673803494611979039124, −5.22750187629811404229061615805, −4.44504999621546393719931019827, −3.49654225242110558210866939024, −2.20790403648055201097741526190, −1.42619831889698812285876070632,
1.05823230756900400081178821362, 1.88207067806001700022014619818, 2.83894326564009810589623182077, 3.77270947372665121411073484189, 4.77841923910192417553390993062, 5.76629235762957487939338578268, 6.09595269087838204457281371345, 6.83243436899213889387488441743, 8.238829143916250075424455438102, 8.895845538621149607907330235568