| L(s) = 1 | + 3-s − 1.19·5-s − 7-s + 9-s − 1.32·11-s + 1.19·13-s − 1.19·15-s − 17-s + 2.46·19-s − 21-s + 7.60·23-s − 3.57·25-s + 27-s − 5.27·29-s − 5.77·31-s − 1.32·33-s + 1.19·35-s − 0.130·37-s + 1.19·39-s − 10.7·41-s + 1.22·43-s − 1.19·45-s − 5.77·47-s + 49-s − 51-s + 11.7·53-s + 1.57·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.533·5-s − 0.377·7-s + 0.333·9-s − 0.398·11-s + 0.330·13-s − 0.307·15-s − 0.242·17-s + 0.565·19-s − 0.218·21-s + 1.58·23-s − 0.715·25-s + 0.192·27-s − 0.979·29-s − 1.03·31-s − 0.230·33-s + 0.201·35-s − 0.0213·37-s + 0.190·39-s − 1.67·41-s + 0.186·43-s − 0.177·45-s − 0.841·47-s + 0.142·49-s − 0.140·51-s + 1.61·53-s + 0.212·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 1.19T + 5T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + 0.130T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2.37T + 61T^{2} \) |
| 67 | \( 1 + 0.0987T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 7.91T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 0.0987T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.68401636255714526684987768137, −7.23886328203842606078216758444, −6.50082888628619179844688036152, −5.50695682596439677162198862095, −4.87818340239453460298254350626, −3.79415477695837148687793533287, −3.38817027220226098726354608736, −2.45695080934908810246302629940, −1.39028717972315563316528764324, 0,
1.39028717972315563316528764324, 2.45695080934908810246302629940, 3.38817027220226098726354608736, 3.79415477695837148687793533287, 4.87818340239453460298254350626, 5.50695682596439677162198862095, 6.50082888628619179844688036152, 7.23886328203842606078216758444, 7.68401636255714526684987768137
