| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 2·11-s − 12-s + 2·13-s + 2·14-s + 16-s + 4·17-s + 18-s − 6·19-s − 2·21-s − 2·22-s − 6·23-s − 24-s + 2·26-s − 27-s + 2·28-s − 10·29-s − 8·31-s + 32-s + 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s − 0.436·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.377·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.75526673158109305683412197744, −6.83473836412740215419986585614, −5.97366017356180364811843248050, −5.59265231439539545281211362327, −4.86522577130028341175536387670, −4.05526722155200952787572765691, −3.46577367428836736206184483608, −2.17149216818055284210626308543, −1.56187629505160408301943587531, 0,
1.56187629505160408301943587531, 2.17149216818055284210626308543, 3.46577367428836736206184483608, 4.05526722155200952787572765691, 4.86522577130028341175536387670, 5.59265231439539545281211362327, 5.97366017356180364811843248050, 6.83473836412740215419986585614, 7.75526673158109305683412197744
