L(s) = 1 | − 7-s − 1.41i·11-s + 13-s + 1.41i·17-s + 19-s − 1.41i·23-s − 1.41i·29-s + 31-s + 43-s − 1.41i·47-s + 1.41i·59-s + 61-s − 67-s + 1.41i·77-s + 1.41i·83-s + ⋯ |
L(s) = 1 | − 7-s − 1.41i·11-s + 13-s + 1.41i·17-s + 19-s − 1.41i·23-s − 1.41i·29-s + 31-s + 43-s − 1.41i·47-s + 1.41i·59-s + 61-s − 67-s + 1.41i·77-s + 1.41i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.059826383\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.059826383\) |
\(L(1)\) |
|
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 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.341590092977878629828543916595, −8.488175431717945062313005350818, −8.102257043226013423307724383503, −6.79555112695010867992420096465, −6.12224649955494156101339490121, −5.67741807178035927669926686561, −4.19746307343099414941982481620, −3.50745555893537077493993703579, −2.60133102228583573717590057644, −0.927879736999085279198485373471,
1.35574886675700468075506851066, 2.79957114848715855936262809137, 3.54627576406584498228687204239, 4.67081895796964342886463253640, 5.47907828137923836687614048949, 6.47279737514731729846813250396, 7.17833273073807590623599083198, 7.77444311268645063685427607118, 9.054463240882008076704081073943, 9.511175960404781618649421398952