| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 216·6-s − 1.30e3·7-s − 512·8-s + 729·9-s − 5.45e3·11-s + 1.72e3·12-s − 5.23e3·13-s + 1.04e4·14-s + 4.09e3·16-s − 4.87e3·17-s − 5.83e3·18-s + 3.27e4·19-s − 3.53e4·21-s + 4.36e4·22-s + 8.34e4·23-s − 1.38e4·24-s + 4.19e4·26-s + 1.96e4·27-s − 8.38e4·28-s − 7.34e4·29-s + 2.44e5·31-s − 3.27e4·32-s − 1.47e5·33-s + 3.89e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.44·7-s − 0.353·8-s + 0.333·9-s − 1.23·11-s + 0.288·12-s − 0.661·13-s + 1.02·14-s + 0.250·16-s − 0.240·17-s − 0.235·18-s + 1.09·19-s − 0.833·21-s + 0.874·22-s + 1.43·23-s − 0.204·24-s + 0.467·26-s + 0.192·27-s − 0.721·28-s − 0.558·29-s + 1.47·31-s − 0.176·32-s − 0.713·33-s + 0.170·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.148251430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148251430\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.30e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.87e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.34e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.44e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.25e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.01e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.22e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.32e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.63e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.09e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.76e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.19e6T + 8.07e13T^{2} \) |
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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
−11.60322915362813590154605305350, −10.21552257058038959263009157012, −9.730895149221286055147911380860, −8.684261031812378004344921641304, −7.53119859192018395936214215869, −6.68601550170479490065799312327, −5.19678783447288911808227169820, −3.29857096706894911872759428991, −2.49436493380797816953037578642, −0.63087753497333766428734834042,
0.63087753497333766428734834042, 2.49436493380797816953037578642, 3.29857096706894911872759428991, 5.19678783447288911808227169820, 6.68601550170479490065799312327, 7.53119859192018395936214215869, 8.684261031812378004344921641304, 9.730895149221286055147911380860, 10.21552257058038959263009157012, 11.60322915362813590154605305350
