| L(s) = 1 | + i·2-s − i·3-s − 4-s + 2.66i·5-s + 6-s − i·8-s − 9-s − 2.66·10-s + (−2.85 + 1.68i)11-s + i·12-s − 2.50·13-s + 2.66·15-s + 16-s − 2.53·17-s − i·18-s + 1.39·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.19i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.843·10-s + (−0.860 + 0.509i)11-s + 0.288i·12-s − 0.693·13-s + 0.688·15-s + 0.250·16-s − 0.614·17-s − 0.235i·18-s + 0.319·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6742194501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6742194501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.85 - 1.68i)T \) |
| good | 5 | \( 1 - 2.66iT - 5T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 + 6.51iT - 29T^{2} \) |
| 31 | \( 1 + 4.05iT - 31T^{2} \) |
| 37 | \( 1 - 2.54T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 + 5.62iT - 43T^{2} \) |
| 47 | \( 1 - 2.76iT - 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 + 4.91iT - 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 8.30iT - 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.03iT - 97T^{2} \) |
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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
−8.084096945856943183501195005657, −7.77219921546963138942577665417, −7.03310480275423751051775480458, −6.48192411880996709177594437261, −5.73726664000874611169366318272, −4.85523949679791229223293621203, −3.89376882086818481708948364523, −2.73806628089032090356671156649, −2.12780876787524434658561246681, −0.23546326788556146775003823381,
0.996843467668158931118491985547, 2.25195468027058826197531084666, 3.17844270001028149514308520543, 4.12282935202658233248124191164, 5.01648213140676844572154242977, 5.20364223147652982645107797060, 6.32157496148588692729891019630, 7.54743331526479341779134155267, 8.297136334383979201121742593155, 8.875069824046190595969057632451
