| L(s) = 1 | + 3-s − 2·4-s − 2·7-s + 9-s + 2·11-s − 2·12-s − 13-s + 4·16-s + 2·17-s − 6·19-s − 2·21-s + 7·23-s + 27-s + 4·28-s − 6·29-s − 6·31-s + 2·33-s − 2·36-s + 4·37-s − 39-s − 7·43-s − 4·44-s + 47-s + 4·48-s − 3·49-s + 2·51-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 0.277·13-s + 16-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 1.45·23-s + 0.192·27-s + 0.755·28-s − 1.11·29-s − 1.07·31-s + 0.348·33-s − 1/3·36-s + 0.657·37-s − 0.160·39-s − 1.06·43-s − 0.603·44-s + 0.145·47-s + 0.577·48-s − 3/7·49-s + 0.280·51-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−8.480319989701192377417583884127, −7.47030934753134433393380603167, −6.84293986853911981148392599267, −5.89628548169783633830818463360, −5.09631436477914448059726013483, −4.14032729043888799898073486115, −3.61253710574935572257674192547, −2.69054419267197679866241565947, −1.39443068139752574368722768823, 0,
1.39443068139752574368722768823, 2.69054419267197679866241565947, 3.61253710574935572257674192547, 4.14032729043888799898073486115, 5.09631436477914448059726013483, 5.89628548169783633830818463360, 6.84293986853911981148392599267, 7.47030934753134433393380603167, 8.480319989701192377417583884127
