| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s + 16-s + 2.82·17-s − 3·18-s − 4.24·19-s − 6·23-s − 5·25-s − 6·29-s + 4.24·31-s + 32-s + 2.82·34-s − 3·36-s + 37-s − 4.24·38-s − 7.07·41-s − 4·43-s − 6·46-s + 2.82·47-s − 5·50-s − 6·58-s + 1.41·59-s + 2.82·61-s + 4.24·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 9-s + 0.250·16-s + 0.685·17-s − 0.707·18-s − 0.973·19-s − 1.25·23-s − 25-s − 1.11·29-s + 0.762·31-s + 0.176·32-s + 0.485·34-s − 0.5·36-s + 0.164·37-s − 0.688·38-s − 1.10·41-s − 0.609·43-s − 0.884·46-s + 0.412·47-s − 0.707·50-s − 0.787·58-s + 0.184·59-s + 0.362·61-s + 0.538·62-s + 0.125·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.135690249933689692712030383394, −7.44341221500459362652491884781, −6.43174950881267442675402629657, −5.87876252389419456870462642474, −5.25774638355562507268228364354, −4.23546773353991904414238491024, −3.56095299368233930503112516588, −2.61937931691366260674987407754, −1.73254520447919951104038560393, 0,
1.73254520447919951104038560393, 2.61937931691366260674987407754, 3.56095299368233930503112516588, 4.23546773353991904414238491024, 5.25774638355562507268228364354, 5.87876252389419456870462642474, 6.43174950881267442675402629657, 7.44341221500459362652491884781, 8.135690249933689692712030383394
