L(s) = 1 | − 835.·3-s + 4.09e3·4-s + 1.51e5·7-s + 1.66e5·9-s + 2.91e6·11-s − 3.42e6·12-s + 1.67e7·16-s + 4.76e7·17-s − 1.26e8·21-s − 2.31e8·23-s + 2.44e8·25-s + 3.04e8·27-s + 6.22e8·28-s + 7.35e7·29-s − 1.48e9·31-s − 2.43e9·33-s + 6.82e8·36-s + 1.55e9·37-s − 9.47e9·41-s + 1.19e10·44-s − 1.40e10·48-s + 9.22e9·49-s − 3.97e10·51-s + 8.40e10·59-s + 5.36e9·61-s + 2.53e10·63-s + 6.87e10·64-s + ⋯ |
L(s) = 1 | − 1.14·3-s + 4-s + 1.29·7-s + 0.313·9-s + 1.64·11-s − 1.14·12-s + 16-s + 1.97·17-s − 1.47·21-s − 1.56·23-s + 25-s + 0.786·27-s + 1.29·28-s + 0.123·29-s − 1.67·31-s − 1.88·33-s + 0.313·36-s + 0.605·37-s − 1.99·41-s + 1.64·44-s − 1.14·48-s + 0.666·49-s − 2.26·51-s + 1.99·59-s + 0.104·61-s + 0.404·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.665566759\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.665566759\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 - 3.26e11T \) |
good | 2 | \( 1 - 4.09e3T^{2} \) |
| 3 | \( 1 + 835.T + 5.31e5T^{2} \) |
| 5 | \( 1 - 2.44e8T^{2} \) |
| 7 | \( 1 - 1.51e5T + 1.38e10T^{2} \) |
| 11 | \( 1 - 2.91e6T + 3.13e12T^{2} \) |
| 13 | \( 1 - 2.32e13T^{2} \) |
| 17 | \( 1 - 4.76e7T + 5.82e14T^{2} \) |
| 19 | \( 1 - 2.21e15T^{2} \) |
| 23 | \( 1 + 2.31e8T + 2.19e16T^{2} \) |
| 29 | \( 1 - 7.35e7T + 3.53e17T^{2} \) |
| 31 | \( 1 + 1.48e9T + 7.87e17T^{2} \) |
| 37 | \( 1 - 1.55e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 9.47e9T + 2.25e19T^{2} \) |
| 43 | \( 1 - 3.99e19T^{2} \) |
| 47 | \( 1 - 1.16e20T^{2} \) |
| 53 | \( 1 - 4.91e20T^{2} \) |
| 59 | \( 1 - 8.40e10T + 1.77e21T^{2} \) |
| 61 | \( 1 - 5.36e9T + 2.65e21T^{2} \) |
| 67 | \( 1 - 8.18e21T^{2} \) |
| 71 | \( 1 - 1.64e22T^{2} \) |
| 73 | \( 1 - 2.29e22T^{2} \) |
| 79 | \( 1 - 5.90e22T^{2} \) |
| 89 | \( 1 - 2.46e23T^{2} \) |
| 97 | \( 1 - 6.93e23T^{2} \) |
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
−11.81405687740170523787975394119, −11.09308532006320970152662817772, −10.04106536208030923888578737051, −8.326131608108892350463358363015, −7.14313871505069713934019061293, −6.06462331461534684840056413365, −5.18902716524000367416630308271, −3.63434061434975935541261509677, −1.76134289431743208580118491072, −0.973435885860746539521554841406,
0.973435885860746539521554841406, 1.76134289431743208580118491072, 3.63434061434975935541261509677, 5.18902716524000367416630308271, 6.06462331461534684840056413365, 7.14313871505069713934019061293, 8.326131608108892350463358363015, 10.04106536208030923888578737051, 11.09308532006320970152662817772, 11.81405687740170523787975394119