| L(s) = 1 | − 3-s + 9-s − 2·11-s − 2·13-s − 2·17-s + 4·19-s − 27-s + 2·29-s + 2·31-s + 2·33-s + 6·37-s + 2·39-s − 10·41-s + 8·43-s − 12·47-s − 7·49-s + 2·51-s − 8·53-s − 4·57-s + 10·59-s + 10·61-s − 8·67-s − 4·73-s + 10·79-s + 81-s + 4·83-s − 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.986·37-s + 0.320·39-s − 1.56·41-s + 1.21·43-s − 1.75·47-s − 49-s + 0.280·51-s − 1.09·53-s − 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.977·67-s − 0.468·73-s + 1.12·79-s + 1/9·81-s + 0.439·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214051673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214051673\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.73948693199723766946197769109, −6.87430124221793133067987911226, −6.42265653144892077428399070792, −5.52474899766298162067050598442, −4.99595232757021996842582520978, −4.39698407746392187239328445291, −3.38202785690311430784338902644, −2.63000674266517061886996492351, −1.66138520918366811772895303941, −0.54103120601857784299951587112,
0.54103120601857784299951587112, 1.66138520918366811772895303941, 2.63000674266517061886996492351, 3.38202785690311430784338902644, 4.39698407746392187239328445291, 4.99595232757021996842582520978, 5.52474899766298162067050598442, 6.42265653144892077428399070792, 6.87430124221793133067987911226, 7.73948693199723766946197769109
