L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s − 0.999·8-s + (1.5 − 2.59i)9-s + (−1.99 − 3.46i)10-s + (0.5 + 0.866i)11-s + 2·13-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (−1.5 − 2.59i)18-s + (3 − 5.19i)19-s − 3.99·20-s + 0.999·22-s + (−2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.632 − 1.09i)10-s + (0.150 + 0.261i)11-s + 0.554·13-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.353 − 0.612i)18-s + (0.688 − 1.19i)19-s − 0.894·20-s + 0.213·22-s + (−0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.319014439\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.319014439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.728487503692433845343612539992, −8.945605507371630456180816962561, −8.296669659523810594535219946907, −6.87389443856581219564747414437, −5.92343046172389565825877820298, −5.17864934667485903209640128342, −4.34838555163970661450082339628, −3.35340507671052236674984709731, −1.75371800366853408060703037084, −1.01892137604708122341118930824,
1.91472600607395798013754341360, 3.01498951779872565081724709347, 3.94538224276158265367873263286, 5.35072494980282081715338562000, 5.92148293867290485265747582528, 6.83697247476284849727067962289, 7.43508509824019733371962286826, 8.278653977682538368213433648929, 9.533770316473619907996675059074, 10.14135246204694752895246302620