| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 4·10-s − 3·11-s − 2·12-s − 6·13-s + 4·14-s + 8·15-s + 16-s − 6·17-s − 18-s − 8·19-s − 4·20-s + 8·21-s + 3·22-s + 2·24-s + 11·25-s + 6·26-s + 4·27-s − 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.904·11-s − 0.577·12-s − 1.66·13-s + 1.06·14-s + 2.06·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.894·20-s + 1.74·21-s + 0.639·22-s + 0.408·24-s + 11/5·25-s + 1.17·26-s + 0.769·27-s − 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 6883 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−16.92490080182032, −16.73117342303382, −15.83099092646616, −15.49349287107565, −15.28139248632153, −14.51133610305555, −13.30003709112075, −12.75637789826160, −12.38047316399303, −11.93310389210260, −11.25485771374990, −10.77766440232521, −10.36228593444782, −9.683012174033356, −8.824604099072401, −8.418610022075906, −7.548287182073239, −7.075792003488712, −6.582874537101454, −6.031743932647601, −4.839587216457532, −4.663582007406859, −3.558052874210275, −2.944798345952740, −2.036516837104735, 0, 0, 0,
2.036516837104735, 2.944798345952740, 3.558052874210275, 4.663582007406859, 4.839587216457532, 6.031743932647601, 6.582874537101454, 7.075792003488712, 7.548287182073239, 8.418610022075906, 8.824604099072401, 9.683012174033356, 10.36228593444782, 10.77766440232521, 11.25485771374990, 11.93310389210260, 12.38047316399303, 12.75637789826160, 13.30003709112075, 14.51133610305555, 15.28139248632153, 15.49349287107565, 15.83099092646616, 16.73117342303382, 16.92490080182032
