| L(s) = 1 | + 2-s − 3-s + 4-s − 4·5-s − 6-s + 3·7-s + 8-s + 9-s − 4·10-s − 2·11-s − 12-s − 2·13-s + 3·14-s + 4·15-s + 16-s − 2·17-s + 18-s − 7·19-s − 4·20-s − 3·21-s − 2·22-s − 6·23-s − 24-s + 11·25-s − 2·26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.60·19-s − 0.894·20-s − 0.654·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.36828839553079636898912742025, −8.673436493776185544675623176808, −7.933830349183438276877648463405, −7.35487302900335587611614440654, −6.31957012190157123135096363551, −5.02189903682911174148969978566, −4.49721695073094645340521851249, −3.67033451303920978757956165896, −2.08991428980413846681790037952, 0,
2.08991428980413846681790037952, 3.67033451303920978757956165896, 4.49721695073094645340521851249, 5.02189903682911174148969978566, 6.31957012190157123135096363551, 7.35487302900335587611614440654, 7.933830349183438276877648463405, 8.673436493776185544675623176808, 10.36828839553079636898912742025
