| L(s) = 1 | + 2.78·3-s + 5-s − 3.76·7-s + 4.76·9-s − 1.15·11-s − 1.63·13-s + 2.78·15-s − 1.38·17-s − 19-s − 10.4·21-s − 7.76·23-s + 25-s + 4.92·27-s − 0.651·29-s + 4.31·31-s − 3.22·33-s − 3.76·35-s − 3.02·37-s − 4.54·39-s + 9.84·41-s − 9.95·43-s + 4.76·45-s − 3.62·47-s + 7.18·49-s − 3.87·51-s − 5.20·53-s − 1.15·55-s + ⋯ |
| L(s) = 1 | + 1.60·3-s + 0.447·5-s − 1.42·7-s + 1.58·9-s − 0.348·11-s − 0.452·13-s + 0.719·15-s − 0.337·17-s − 0.229·19-s − 2.29·21-s − 1.61·23-s + 0.200·25-s + 0.947·27-s − 0.121·29-s + 0.774·31-s − 0.560·33-s − 0.636·35-s − 0.496·37-s − 0.727·39-s + 1.53·41-s − 1.51·43-s + 0.710·45-s − 0.528·47-s + 1.02·49-s − 0.542·51-s − 0.714·53-s − 0.155·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 23 | \( 1 + 7.76T + 23T^{2} \) |
| 29 | \( 1 + 0.651T + 29T^{2} \) |
| 31 | \( 1 - 4.31T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 1.15T + 61T^{2} \) |
| 67 | \( 1 + 1.99T + 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 0.0412T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 + 0.794T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.890811823829967399006063930824, −7.06918310907952698151310534597, −6.43452268590488028174795377666, −5.72536861475087711181557285397, −4.56320769957724924864170024662, −3.82078395620635332309251242258, −3.05063158920383971719622372592, −2.53065891155452846993551301706, −1.71048883201835333828131723079, 0,
1.71048883201835333828131723079, 2.53065891155452846993551301706, 3.05063158920383971719622372592, 3.82078395620635332309251242258, 4.56320769957724924864170024662, 5.72536861475087711181557285397, 6.43452268590488028174795377666, 7.06918310907952698151310534597, 7.890811823829967399006063930824
