| L(s) = 1 | − 21.9·2-s + 9.99·3-s + 354.·4-s + 125·5-s − 219.·6-s + 343·7-s − 4.98e3·8-s − 2.08e3·9-s − 2.74e3·10-s + 4.69e3·11-s + 3.54e3·12-s − 1.37e3·13-s − 7.53e3·14-s + 1.24e3·15-s + 6.40e4·16-s − 2.08e4·17-s + 4.58e4·18-s + 5.23e4·19-s + 4.43e4·20-s + 3.42e3·21-s − 1.03e5·22-s + 2.27e4·23-s − 4.98e4·24-s + 1.56e4·25-s + 3.01e4·26-s − 4.27e4·27-s + 1.21e5·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.213·3-s + 2.77·4-s + 0.447·5-s − 0.415·6-s + 0.377·7-s − 3.44·8-s − 0.954·9-s − 0.868·10-s + 1.06·11-s + 0.592·12-s − 0.173·13-s − 0.734·14-s + 0.0956·15-s + 3.91·16-s − 1.02·17-s + 1.85·18-s + 1.75·19-s + 1.23·20-s + 0.0808·21-s − 2.06·22-s + 0.390·23-s − 0.735·24-s + 0.199·25-s + 0.336·26-s − 0.417·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8665127486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8665127486\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 21.9T + 128T^{2} \) |
| 3 | \( 1 - 9.99T + 2.18e3T^{2} \) |
| 11 | \( 1 - 4.69e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.37e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.08e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.66e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.48e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.16e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.19e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.35e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.99e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.23e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.22e6T + 8.07e13T^{2} \) |
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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
−15.38411339580587681766466164097, −14.12831285654820229109915260418, −11.86278876517181200316418612661, −11.04544315610457071982076010096, −9.554144538873110269560233225097, −8.839499909479441855205320341275, −7.52196948460515205212032547527, −6.11089241668153167180422343908, −2.61401530443273540639120601509, −1.00390428722614006707648473964,
1.00390428722614006707648473964, 2.61401530443273540639120601509, 6.11089241668153167180422343908, 7.52196948460515205212032547527, 8.839499909479441855205320341275, 9.554144538873110269560233225097, 11.04544315610457071982076010096, 11.86278876517181200316418612661, 14.12831285654820229109915260418, 15.38411339580587681766466164097
