| L(s) = 1 | − 8.49i·3-s − 59.7·5-s − 483. i·7-s + 656.·9-s + 1.41e3i·11-s − 3.45e3·13-s + 507. i·15-s − 3.05e3·17-s − 968. i·19-s − 4.10e3·21-s − 3.31e3i·23-s − 1.20e4·25-s − 1.17e4i·27-s + 2.63e4·29-s − 2.71e4i·31-s + ⋯ |
| L(s) = 1 | − 0.314i·3-s − 0.477·5-s − 1.40i·7-s + 0.901·9-s + 1.06i·11-s − 1.57·13-s + 0.150i·15-s − 0.622·17-s − 0.141i·19-s − 0.443·21-s − 0.272i·23-s − 0.771·25-s − 0.598i·27-s + 1.08·29-s − 0.909i·31-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
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6619678205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6619678205\) |
| \(L(4)\) |
|
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 | \( 1 + 8.49iT - 729T^{2} \) |
| 5 | \( 1 + 59.7T + 1.56e4T^{2} \) |
| 7 | \( 1 + 483. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.41e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.45e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.05e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 968. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.31e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.63e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.71e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.60e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 6.86e3T + 4.75e9T^{2} \) |
| 43 | \( 1 - 9.28e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 8.66e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.28e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.89e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.19e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.96e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.39e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.64e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 8.02e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 5.41e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.10e6T + 8.32e11T^{2} \) |
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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
−11.19059112927253221308322452750, −10.08796027973584745503313874899, −9.620570043283287714624339095442, −7.81318090227848965734095189096, −7.39224353805116516728348854803, −6.56580306210683017506124312759, −4.58458813689210568216479422108, −4.25027081322391383865058686915, −2.45255899310010222121206104282, −1.07629607635726518056824546255,
0.18283820631947604492357941034, 2.02062137185212549035992421583, 3.20509868231752500009451107893, 4.56236932334046916450097739797, 5.50123379763009442142974429384, 6.72434613361066284725186936077, 7.88261120958260476744443659002, 8.850810430593528350366435186073, 9.688966706470480298200164696962, 10.70744305060492086752222844815
