| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s − 2·9-s − 3·10-s + 2·11-s + 12-s + 14-s − 3·15-s + 16-s + 17-s − 2·18-s − 19-s − 3·20-s + 21-s + 2·22-s − 4·23-s + 24-s + 4·25-s − 5·27-s + 28-s + 2·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s − 0.229·19-s − 0.670·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 4/5·25-s − 0.962·27-s + 0.188·28-s + 0.371·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.88757284150136766732477664550, −7.01637387521731614909925436851, −6.23985117494634009733502774393, −5.46230235312862722482784714917, −4.55001846829100772490283220371, −3.96158001504007113865856765064, −3.36985240269987053144044187851, −2.58564575735255395386800609962, −1.50081708010322453180627524892, 0,
1.50081708010322453180627524892, 2.58564575735255395386800609962, 3.36985240269987053144044187851, 3.96158001504007113865856765064, 4.55001846829100772490283220371, 5.46230235312862722482784714917, 6.23985117494634009733502774393, 7.01637387521731614909925436851, 7.88757284150136766732477664550
