| L(s) = 1 | + i·7-s − 3·11-s − i·13-s + 6i·17-s + 4·19-s + 3i·23-s + 3·29-s + 5·31-s − 2i·37-s − 3·41-s − i·43-s − 9i·47-s + 6·49-s + 6i·53-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s − 0.904·11-s − 0.277i·13-s + 1.45i·17-s + 0.917·19-s + 0.625i·23-s + 0.557·29-s + 0.898·31-s − 0.328i·37-s − 0.468·41-s − 0.152i·43-s − 1.31i·47-s + 0.857·49-s + 0.824i·53-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244589423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244589423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 11iT - 97T^{2} \) |
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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
−8.020447453028799337353840311442, −7.49594367466110558551585549445, −6.63623637002773688387976756038, −5.84958523982413212493460525913, −5.38906709466205995081866338723, −4.58830202383528325011295839015, −3.68124625247560916230419077705, −2.95317231329457203333309748326, −2.11576947482009928898836048946, −1.09345759444443759037937674993,
0.31877350296695350286436707115, 1.33398030643492320724505886367, 2.64089445190707309823216575437, 3.00256914967711954345464833986, 4.14398147801413692709452445638, 4.86665600813687611258936859091, 5.34423240662575534142109088982, 6.34053647716306770695072107889, 6.92856319602506406522801114572, 7.66826553467036677639334793517
