| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s + 2·11-s + 12-s − 5·13-s − 3·15-s + 16-s − 4·17-s + 18-s + 6·19-s − 3·20-s + 2·22-s + 23-s + 24-s + 4·25-s − 5·26-s + 27-s + 29-s − 3·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.603·11-s + 0.288·12-s − 1.38·13-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.670·20-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.980·26-s + 0.192·27-s + 0.185·29-s − 0.547·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.55712127763544529070179431506, −7.08796673769646943071617478761, −6.35278257700833409284590441990, −5.21237626129869733867643547648, −4.62058131481960057747935315807, −3.99697947156727681197640790078, −3.26059602197126681289746323989, −2.61570289792596412409665058048, −1.47076845530614876821757293135, 0,
1.47076845530614876821757293135, 2.61570289792596412409665058048, 3.26059602197126681289746323989, 3.99697947156727681197640790078, 4.62058131481960057747935315807, 5.21237626129869733867643547648, 6.35278257700833409284590441990, 7.08796673769646943071617478761, 7.55712127763544529070179431506
