| L(s) = 1 | + 0.347·2-s − 0.879·4-s − 1.87·5-s − 0.652·8-s + 9-s − 0.652·10-s + 0.652·16-s + 0.347·18-s + 1.53·19-s + 1.65·20-s + 2.53·25-s + 31-s + 0.879·32-s − 0.879·36-s + 0.532·38-s + 1.22·40-s + 0.347·41-s − 1.87·45-s − 47-s + 0.879·50-s + 0.347·59-s + 0.347·62-s − 0.347·64-s − 67-s + 1.53·71-s − 0.652·72-s − 1.34·76-s + ⋯ |
| L(s) = 1 | + 0.347·2-s − 0.879·4-s − 1.87·5-s − 0.652·8-s + 9-s − 0.652·10-s + 0.652·16-s + 0.347·18-s + 1.53·19-s + 1.65·20-s + 2.53·25-s + 31-s + 0.879·32-s − 0.879·36-s + 0.532·38-s + 1.22·40-s + 0.347·41-s − 1.87·45-s − 47-s + 0.879·50-s + 0.347·59-s + 0.347·62-s − 0.347·64-s − 67-s + 1.53·71-s − 0.652·72-s − 1.34·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7914123373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7914123373\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.347T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.347T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - 1.53T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.53T + T^{2} \) |
| 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
−9.616230876635770483963441151255, −8.777674307805174207411100966756, −7.924117616702260715898240547545, −7.49306924840174705775518066469, −6.52624915984847684343292951118, −5.15801449213831719551844462716, −4.53887474819043531732364742425, −3.80397147865340992980554819120, −3.12377792501494331130167278924, −0.925997377301685176409720558602,
0.925997377301685176409720558602, 3.12377792501494331130167278924, 3.80397147865340992980554819120, 4.53887474819043531732364742425, 5.15801449213831719551844462716, 6.52624915984847684343292951118, 7.49306924840174705775518066469, 7.924117616702260715898240547545, 8.777674307805174207411100966756, 9.616230876635770483963441151255
