L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 3·11-s + 5·13-s + 16-s − 6·17-s − 19-s + 20-s + 3·22-s − 3·23-s + 25-s − 5·26-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 11·37-s + 38-s − 40-s − 3·41-s − 10·43-s − 3·44-s + 3·46-s − 3·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s − 0.980·26-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.468·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.179612029398429829609125634718, −7.39034799988300208199878423892, −6.35258858809397583856109163406, −6.18331139266297557267541271099, −5.07518526534012859577823681630, −4.22810552630422532228110095479, −3.13992700813965508494041609090, −2.28487132600936674264447181601, −1.38445309591670219666322117745, 0,
1.38445309591670219666322117745, 2.28487132600936674264447181601, 3.13992700813965508494041609090, 4.22810552630422532228110095479, 5.07518526534012859577823681630, 6.18331139266297557267541271099, 6.35258858809397583856109163406, 7.39034799988300208199878423892, 8.179612029398429829609125634718