L(s) = 1 | − 81·3-s − 625·5-s + 4.01e3·7-s + 6.56e3·9-s − 8.48e4·11-s + 1.19e5·13-s + 5.06e4·15-s + 1.16e5·17-s + 2.34e5·19-s − 3.24e5·21-s − 2.34e6·23-s + 3.90e5·25-s − 5.31e5·27-s − 4.64e5·29-s + 5.11e6·31-s + 6.87e6·33-s − 2.50e6·35-s + 8.69e6·37-s − 9.67e6·39-s − 9.05e6·41-s − 8.63e6·43-s − 4.10e6·45-s − 3.31e7·47-s − 2.42e7·49-s − 9.47e6·51-s − 6.41e7·53-s + 5.30e7·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.631·7-s + 0.333·9-s − 1.74·11-s + 1.15·13-s + 0.258·15-s + 0.339·17-s + 0.413·19-s − 0.364·21-s − 1.74·23-s + 0.200·25-s − 0.192·27-s − 0.121·29-s + 0.995·31-s + 1.00·33-s − 0.282·35-s + 0.762·37-s − 0.669·39-s − 0.500·41-s − 0.385·43-s − 0.149·45-s − 0.990·47-s − 0.601·49-s − 0.196·51-s − 1.11·53-s + 0.781·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.250067177\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250067177\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| 5 | \( 1 + 625T \) |
good | 7 | \( 1 - 4.01e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.48e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.19e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.34e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.64e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.11e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.69e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.63e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.31e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.41e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.49e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.54e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.06e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.17e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.32e9T + 7.60e17T^{2} \) |
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
−10.66683304307045214018557953336, −9.779814413332066795473311667615, −8.104019346006165527132416540627, −7.929227391751733411359698196316, −6.39788060244603872973061895633, −5.42143717186757852901116049022, −4.50286353104763628960535066187, −3.23076233337456084916915390970, −1.80887010508796401555275439371, −0.53087056877413037517227093137,
0.53087056877413037517227093137, 1.80887010508796401555275439371, 3.23076233337456084916915390970, 4.50286353104763628960535066187, 5.42143717186757852901116049022, 6.39788060244603872973061895633, 7.929227391751733411359698196316, 8.104019346006165527132416540627, 9.779814413332066795473311667615, 10.66683304307045214018557953336