L(s) = 1 | + (1 + i)3-s + (−1 + i)7-s + i·9-s − 2·21-s + (1 + i)23-s − 2i·29-s + (−1 − i)43-s + (1 − i)47-s − i·49-s + (−1 − i)63-s + (−1 + i)67-s + 2i·69-s + 81-s + (−1 − i)83-s + (2 − 2i)87-s + ⋯ |
L(s) = 1 | + (1 + i)3-s + (−1 + i)7-s + i·9-s − 2·21-s + (1 + i)23-s − 2i·29-s + (−1 − i)43-s + (1 − i)47-s − i·49-s + (−1 − i)63-s + (−1 + i)67-s + 2i·69-s + 81-s + (−1 − i)83-s + (2 − 2i)87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.203153927\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203153927\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.25418350284940333018242297933, −9.712318834154996844364446445339, −9.035576405082033067896923317804, −8.472426697580318999022771052851, −7.33705878273929921372252718273, −6.18178692110613313626959723916, −5.27022331743516471751462930084, −4.06560687717947572477017256657, −3.21439990882608355955517968670, −2.37520475548774116647543465138,
1.27596552012151008585594330858, 2.77337933298861100698574492004, 3.48879543380692432366847665689, 4.78912913688693580623298635546, 6.32662121816747505270602289693, 6.99530368165946003159275129257, 7.57101883992844295644297705453, 8.602158146563770624393973871475, 9.235042093885249695731507453882, 10.27436645597235270369076056873