| L(s) = 1 | + 8·2-s − 19.0·3-s + 64·4-s + 319.·5-s − 152.·6-s − 1.29e3·7-s + 512·8-s − 1.82e3·9-s + 2.55e3·10-s + 5.36e3·11-s − 1.21e3·12-s + 1.88e3·13-s − 1.03e4·14-s − 6.08e3·15-s + 4.09e3·16-s − 1.78e3·17-s − 1.45e4·18-s − 3.24e4·19-s + 2.04e4·20-s + 2.46e4·21-s + 4.28e4·22-s + 4.57e4·23-s − 9.75e3·24-s + 2.37e4·25-s + 1.50e4·26-s + 7.64e4·27-s − 8.26e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.407·3-s + 0.5·4-s + 1.14·5-s − 0.288·6-s − 1.42·7-s + 0.353·8-s − 0.833·9-s + 0.807·10-s + 1.21·11-s − 0.203·12-s + 0.237·13-s − 1.00·14-s − 0.465·15-s + 0.250·16-s − 0.0883·17-s − 0.589·18-s − 1.08·19-s + 0.570·20-s + 0.579·21-s + 0.858·22-s + 0.784·23-s − 0.144·24-s + 0.303·25-s + 0.167·26-s + 0.747·27-s − 0.711·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 + 19.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 319.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.29e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.36e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.88e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.01e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.93e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.35e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.41e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.64e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.90e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.42e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.16e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.76e6T + 8.07e13T^{2} \) |
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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
−9.394625641796421373719006181565, −8.551101947914979701455202661573, −6.84209267072664409598067838731, −6.27207496983061366711120498900, −5.86190243773638856056670402868, −4.63219981926432132653450629151, −3.43319138671183374778856277303, −2.59179099878345394664280960881, −1.36620458845618597343628635512, 0,
1.36620458845618597343628635512, 2.59179099878345394664280960881, 3.43319138671183374778856277303, 4.63219981926432132653450629151, 5.86190243773638856056670402868, 6.27207496983061366711120498900, 6.84209267072664409598067838731, 8.551101947914979701455202661573, 9.394625641796421373719006181565
