| L(s) = 1 | − 2.63·2-s + 27·3-s − 121.·4-s + 254.·5-s − 71.0·6-s − 1.40e3·7-s + 655.·8-s + 729·9-s − 670.·10-s − 3.26e3·12-s + 1.38e4·13-s + 3.69e3·14-s + 6.87e3·15-s + 1.37e4·16-s + 1.70e4·17-s − 1.91e3·18-s − 1.55e4·19-s − 3.08e4·20-s − 3.78e4·21-s − 5.89e4·23-s + 1.77e4·24-s − 1.32e4·25-s − 3.64e4·26-s + 1.96e4·27-s + 1.69e5·28-s − 3.08e4·29-s − 1.81e4·30-s + ⋯ |
| L(s) = 1 | − 0.232·2-s + 0.577·3-s − 0.945·4-s + 0.911·5-s − 0.134·6-s − 1.54·7-s + 0.452·8-s + 0.333·9-s − 0.212·10-s − 0.546·12-s + 1.75·13-s + 0.359·14-s + 0.526·15-s + 0.840·16-s + 0.841·17-s − 0.0775·18-s − 0.519·19-s − 0.861·20-s − 0.892·21-s − 1.00·23-s + 0.261·24-s − 0.169·25-s − 0.407·26-s + 0.192·27-s + 1.46·28-s − 0.234·29-s − 0.122·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.847087772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847087772\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 128T^{2} \) |
| 5 | \( 1 - 254.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.40e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 1.38e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.89e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.08e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.94e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.04e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.82e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.76e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.25e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.57e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00940004781029497314564061689, −9.252498529717313391713792087532, −8.739130898800700154829004742483, −7.58125057833717488671059979508, −6.21729235479718120233249112720, −5.66818878997053419082380590413, −3.96151147286320475189843756349, −3.37043554430482354627082168670, −1.88177294572135039219714888053, −0.65359969783680061688184183840,
0.65359969783680061688184183840, 1.88177294572135039219714888053, 3.37043554430482354627082168670, 3.96151147286320475189843756349, 5.66818878997053419082380590413, 6.21729235479718120233249112720, 7.58125057833717488671059979508, 8.739130898800700154829004742483, 9.252498529717313391713792087532, 10.00940004781029497314564061689
