| L(s) = 1 | + (−102. − 102. i)7-s − 185. i·11-s + (−729. + 729. i)13-s + (−1.10e3 + 1.10e3i)17-s − 2.03e3i·19-s + (1.54e3 + 1.54e3i)23-s − 6.65e3·29-s − 1.25e3·31-s + (2.69e3 + 2.69e3i)37-s + 1.85e4i·41-s + (−2.44e3 + 2.44e3i)43-s + (1.40e4 − 1.40e4i)47-s + 4.14e3i·49-s + (1.37e4 + 1.37e4i)53-s + 2.89e4·59-s + ⋯ |
| L(s) = 1 | + (−0.789 − 0.789i)7-s − 0.462i·11-s + (−1.19 + 1.19i)13-s + (−0.923 + 0.923i)17-s − 1.29i·19-s + (0.607 + 0.607i)23-s − 1.46·29-s − 0.233·31-s + (0.323 + 0.323i)37-s + 1.72i·41-s + (−0.201 + 0.201i)43-s + (0.927 − 0.927i)47-s + 0.246i·49-s + (0.674 + 0.674i)53-s + 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.081109943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081109943\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (102. + 102. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 185. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (729. - 729. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.10e3 - 1.10e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.03e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.54e3 - 1.54e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 6.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-2.69e3 - 2.69e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.85e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (2.44e3 - 2.44e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.40e4 + 1.40e4i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.37e4 - 1.37e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 2.89e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.31e3 - 3.31e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.75e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.17e4 - 1.17e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 6.89e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (4.19e4 + 4.19e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.67e4 + 1.67e4i)T + 8.58e9iT^{2} \) |
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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
−9.361746007305357823608012962337, −8.668731126156933279800708521999, −7.32967705189622052107507951116, −6.94083581152028972172486673762, −6.00179695604578513167570026726, −4.76726963605206496645305719629, −4.00786009992243104925039189923, −2.93451437255508087384365271449, −1.80515533750616734314841432681, −0.42029873787510616516611410273,
0.46529318344534840764065275602, 2.15123604289549248421671457577, 2.83985944872935382506702041766, 4.00257970097469725000928255370, 5.22310233861760933071235703094, 5.79173716586140778259341191594, 6.97395687764322899886180771427, 7.58557263810646982940945663334, 8.711723302692184298014434592756, 9.446508582100998717435158314102
