| L(s) = 1 | + 0.347·2-s + 3-s − 0.879·4-s + 0.347·6-s − 1.87·7-s − 0.652·8-s + 9-s + 1.53·11-s − 0.879·12-s + 0.347·13-s − 0.652·14-s + 0.652·16-s + 1.53·17-s + 0.347·18-s − 1.87·21-s + 0.532·22-s − 0.652·24-s + 0.120·26-s + 27-s + 1.65·28-s + 29-s + 0.879·32-s + 1.53·33-s + 0.532·34-s − 0.879·36-s + 0.347·39-s − 41-s + ⋯ |
| L(s) = 1 | + 0.347·2-s + 3-s − 0.879·4-s + 0.347·6-s − 1.87·7-s − 0.652·8-s + 9-s + 1.53·11-s − 0.879·12-s + 0.347·13-s − 0.652·14-s + 0.652·16-s + 1.53·17-s + 0.347·18-s − 1.87·21-s + 0.532·22-s − 0.652·24-s + 0.120·26-s + 27-s + 1.65·28-s + 29-s + 0.879·32-s + 1.53·33-s + 0.532·34-s − 0.879·36-s + 0.347·39-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.501675650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501675650\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.347T + T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 - 0.347T + T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.87T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.347T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.257938083134648147928482854709, −8.691383616737742119879953717882, −7.86389041499769093945870291767, −6.68791974876211866439026023359, −6.34545213277789513529148227185, −5.19004193341438494320000767310, −3.98505075658530088173752754451, −3.54680228661875330699676854351, −2.94212328958274825648708830591, −1.18432516009006482961781076353,
1.18432516009006482961781076353, 2.94212328958274825648708830591, 3.54680228661875330699676854351, 3.98505075658530088173752754451, 5.19004193341438494320000767310, 6.34545213277789513529148227185, 6.68791974876211866439026023359, 7.86389041499769093945870291767, 8.691383616737742119879953717882, 9.257938083134648147928482854709
