| L(s) = 1 | + 3-s + 2·5-s + 9-s − 4·11-s − 13-s + 2·15-s + 17-s + 4·19-s − 25-s + 27-s + 2·29-s − 8·31-s − 4·33-s + 2·37-s − 39-s + 2·41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s + 51-s + 10·53-s − 8·55-s + 4·57-s − 4·59-s − 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.140·51-s + 1.37·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s − 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.97354348298255, −14.32524200286004, −13.86756144104277, −13.59381691833333, −12.84630744858541, −12.68077855410029, −11.91108680160799, −11.28551608154144, −10.59821457899757, −10.23369533268342, −9.703637746912206, −9.202256631420371, −8.752258441185727, −7.947733458070073, −7.515819141452314, −7.147123509279834, −6.200443132405007, −5.664507676254207, −5.276368609168612, −4.549768021220500, −3.822537174132643, −2.997814565122510, −2.593854119500655, −1.897176430948922, −1.161356443259139, 0,
1.161356443259139, 1.897176430948922, 2.593854119500655, 2.997814565122510, 3.822537174132643, 4.549768021220500, 5.276368609168612, 5.664507676254207, 6.200443132405007, 7.147123509279834, 7.515819141452314, 7.947733458070073, 8.752258441185727, 9.202256631420371, 9.703637746912206, 10.23369533268342, 10.59821457899757, 11.28551608154144, 11.91108680160799, 12.68077855410029, 12.84630744858541, 13.59381691833333, 13.86756144104277, 14.32524200286004, 14.97354348298255
