| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s + 3·11-s + 12-s + 14-s + 16-s + 2·18-s − 2·19-s − 21-s − 3·22-s − 3·23-s − 24-s − 5·25-s − 5·27-s − 28-s − 5·31-s − 32-s + 3·33-s − 2·36-s + 7·37-s + 2·38-s − 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.471·18-s − 0.458·19-s − 0.218·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s − 25-s − 0.962·27-s − 0.188·28-s − 0.898·31-s − 0.176·32-s + 0.522·33-s − 1/3·36-s + 1.15·37-s + 0.324·38-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.741970474071750129887812750817, −7.901246002581365567663000678054, −7.31512555904981211148499292732, −6.20675945940118264214195897884, −5.84237101905965821728575679765, −4.36118677889129813821357635938, −3.51081458235067745920013483532, −2.59099936648663070630528278880, −1.59851142262446230512480153730, 0,
1.59851142262446230512480153730, 2.59099936648663070630528278880, 3.51081458235067745920013483532, 4.36118677889129813821357635938, 5.84237101905965821728575679765, 6.20675945940118264214195897884, 7.31512555904981211148499292732, 7.901246002581365567663000678054, 8.741970474071750129887812750817
