L(s) = 1 | − 3-s + 3.46i·7-s − 2·9-s + 3i·11-s + 3.46·13-s + 3i·17-s − i·19-s − 3.46i·21-s + 5·27-s − 10.3i·29-s − 6.92·31-s − 3i·33-s − 10.3·37-s − 3.46·39-s − 9·41-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.30i·7-s − 0.666·9-s + 0.904i·11-s + 0.960·13-s + 0.727i·17-s − 0.229i·19-s − 0.755i·21-s + 0.962·27-s − 1.92i·29-s − 1.24·31-s − 0.522i·33-s − 1.70·37-s − 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5492635091\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5492635091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.750336410761728357350148703566, −8.838296890938927683453964684086, −8.433660514584301051209961802506, −7.34494055708296366960485669212, −6.25440215184288638670466447591, −5.81611767167689992964189465276, −5.02877852907309588390505642347, −3.92632188978250475156374063359, −2.73748813385637224814351707850, −1.71852454456417866311494273298,
0.23276194127581505080245378699, 1.43170118626755737657459609348, 3.22777628827437212835551423331, 3.77067883993671132444663164287, 5.09465194984217825982339193373, 5.64575826171086308817580322910, 6.74771263338492186431256355205, 7.18290325341912050590567027456, 8.445956368622358669672806673299, 8.801918847667939999979090283733