| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 2·13-s − 15-s − 6·17-s − 4·19-s + 21-s + 25-s + 27-s − 2·29-s + 4·31-s + 33-s − 35-s − 10·37-s + 2·39-s − 2·41-s + 12·43-s − 45-s + 4·47-s + 49-s − 6·51-s + 2·53-s − 55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.840·51-s + 0.274·53-s − 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.25603300722904, −13.77878861784942, −13.52476771491055, −12.82872103843515, −12.30793557676397, −11.97851121221622, −11.16361458560826, −10.73706344362340, −10.63284258288671, −9.594093634080604, −9.163142727936275, −8.746986290452221, −8.215028602717508, −7.862414224457742, −7.059776040651083, −6.689852763624945, −6.168242151240261, −5.366271808889745, −4.744349151626505, −4.067922656658981, −3.898843587584562, −3.017293688601165, −2.321717346312295, −1.819617768459405, −0.9592233869659655, 0,
0.9592233869659655, 1.819617768459405, 2.321717346312295, 3.017293688601165, 3.898843587584562, 4.067922656658981, 4.744349151626505, 5.366271808889745, 6.168242151240261, 6.689852763624945, 7.059776040651083, 7.862414224457742, 8.215028602717508, 8.746986290452221, 9.163142727936275, 9.594093634080604, 10.63284258288671, 10.73706344362340, 11.16361458560826, 11.97851121221622, 12.30793557676397, 12.82872103843515, 13.52476771491055, 13.77878861784942, 14.25603300722904
