| L(s) = 1 | + 46i·3-s − 1.38e3·9-s + 2.33e3i·11-s − 1.72e3·17-s − 2.48e3i·19-s − 1.56e4·25-s − 3.02e4i·27-s − 1.07e5·33-s − 1.34e5·41-s + 7.49e4i·43-s + 1.17e5·49-s − 7.93e4i·51-s + 1.14e5·57-s − 3.04e5i·59-s − 5.96e5i·67-s + ⋯ |
| L(s) = 1 | + 1.70i·3-s − 1.90·9-s + 1.75i·11-s − 0.351·17-s − 0.361i·19-s − 25-s − 1.53i·27-s − 2.99·33-s − 1.95·41-s + 0.942i·43-s + 49-s − 0.598i·51-s + 0.616·57-s − 1.48i·59-s − 1.98i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4851977642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4851977642\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 46iT - 729T^{2} \) |
| 5 | \( 1 + 1.56e4T^{2} \) |
| 7 | \( 1 - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.33e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.72e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 2.48e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.48e8T^{2} \) |
| 29 | \( 1 + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.34e5T + 4.75e9T^{2} \) |
| 43 | \( 1 - 7.49e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.04e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.96e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.93e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.78e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.57e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.82e6T + 8.32e11T^{2} \) |
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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
−11.46032832461422437498470920580, −10.49673472560877076287147646902, −9.772046672517087171576322596535, −9.195687696054833093342330133809, −7.954938260543692240732324046305, −6.63452839026957941631420292790, −5.18793932610575599342777760826, −4.52430227267389778495550521223, −3.54476157437883760203065766044, −2.10851925154106419350314046332,
0.12738945172726314096066887433, 1.13690728468161877757145629769, 2.32745868636680488367607960742, 3.55922123814746324165451279920, 5.54493039481602849544264642851, 6.27255232848034694235782283807, 7.24803858976984469675096417288, 8.216778418187485596194885798916, 8.838164377566353037457862448000, 10.45060087747704012913459331370
