L(s) = 1 | + (0.982 + 0.183i)4-s + (−0.404 − 1.42i)7-s + (−0.538 − 1.89i)13-s + (0.932 + 0.361i)16-s + (0.658 + 0.600i)19-s + (−0.602 + 0.798i)25-s + (−0.136 − 1.47i)28-s + (−0.380 + 0.981i)31-s + (1.01 + 1.63i)37-s + (−0.840 + 1.35i)43-s + (−1.00 + 0.623i)49-s + (−0.181 − 1.95i)52-s + (1.12 − 1.48i)61-s + (0.850 + 0.526i)64-s + (−1.91 − 0.544i)67-s + ⋯ |
L(s) = 1 | + (0.982 + 0.183i)4-s + (−0.404 − 1.42i)7-s + (−0.538 − 1.89i)13-s + (0.932 + 0.361i)16-s + (0.658 + 0.600i)19-s + (−0.602 + 0.798i)25-s + (−0.136 − 1.47i)28-s + (−0.380 + 0.981i)31-s + (1.01 + 1.63i)37-s + (−0.840 + 1.35i)43-s + (−1.00 + 0.623i)49-s + (−0.181 − 1.95i)52-s + (1.12 − 1.48i)61-s + (0.850 + 0.526i)64-s + (−1.91 − 0.544i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.178230333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178230333\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + (-0.850 + 0.526i)T \) |
good | 2 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 5 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 7 | \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \) |
| 11 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 13 | \( 1 + (0.538 + 1.89i)T + (-0.850 + 0.526i)T^{2} \) |
| 17 | \( 1 + (-0.273 - 0.961i)T^{2} \) |
| 19 | \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \) |
| 23 | \( 1 + (-0.982 + 0.183i)T^{2} \) |
| 29 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 31 | \( 1 + (0.380 - 0.981i)T + (-0.739 - 0.673i)T^{2} \) |
| 37 | \( 1 + (-1.01 - 1.63i)T + (-0.445 + 0.895i)T^{2} \) |
| 41 | \( 1 + (-0.602 - 0.798i)T^{2} \) |
| 43 | \( 1 + (0.840 - 1.35i)T + (-0.445 - 0.895i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 59 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 1.48i)T + (-0.273 - 0.961i)T^{2} \) |
| 67 | \( 1 + (1.91 + 0.544i)T + (0.850 + 0.526i)T^{2} \) |
| 71 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 73 | \( 1 + (-0.646 - 0.322i)T + (0.602 + 0.798i)T^{2} \) |
| 79 | \( 1 + (-0.243 - 0.489i)T + (-0.602 + 0.798i)T^{2} \) |
| 83 | \( 1 + (-0.850 + 0.526i)T^{2} \) |
| 89 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 97 | \( 1 + (0.726 + 0.961i)T + (-0.273 + 0.961i)T^{2} \) |
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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
−10.17394763133797488506124863842, −9.783852701195162210897998869796, −8.104216446246418142493810947789, −7.67801156681774777784383009377, −6.91855474870432141497861705302, −6.00670114230848600890501084261, −4.98246359182416361549919548656, −3.55256096737886480277565002443, −2.99120654652935979644801935729, −1.28092566477451443098327616928,
2.03167008809096728319065461568, 2.60137446843220795654745994309, 4.04111736000831319067966682926, 5.34691529868382475109355293963, 6.10354223059924382252062242266, 6.86889184651880828296963177723, 7.67406832008584808452935277914, 8.949939035679379301823474717624, 9.386669851776245709172755531877, 10.31229078357398417528650601124