| L(s) = 1 | + 7.35·2-s − 9·3-s + 22.0·4-s − 66.1·6-s + 251.·7-s − 72.9·8-s + 81·9-s + 121·11-s − 198.·12-s − 633.·13-s + 1.85e3·14-s − 1.24e3·16-s + 460.·17-s + 595.·18-s − 1.69e3·19-s − 2.26e3·21-s + 889.·22-s − 3.13e3·23-s + 656.·24-s − 4.66e3·26-s − 729·27-s + 5.56e3·28-s + 223.·29-s − 7.39e3·31-s − 6.80e3·32-s − 1.08e3·33-s + 3.38e3·34-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.689·4-s − 0.750·6-s + 1.94·7-s − 0.403·8-s + 0.333·9-s + 0.301·11-s − 0.398·12-s − 1.04·13-s + 2.52·14-s − 1.21·16-s + 0.386·17-s + 0.433·18-s − 1.07·19-s − 1.12·21-s + 0.391·22-s − 1.23·23-s + 0.232·24-s − 1.35·26-s − 0.192·27-s + 1.34·28-s + 0.0492·29-s − 1.38·31-s − 1.17·32-s − 0.174·33-s + 0.502·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 7.35T + 32T^{2} \) |
| 7 | \( 1 - 251.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 633.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 460.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.69e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.13e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 223.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.93e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.50e4T + 8.58e9T^{2} \) |
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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
−8.997401826351615974945902154762, −7.992316230922769723704960580517, −7.20498580264196947433326088967, −5.99319431655404024742384634043, −5.38874182297044395884066521337, −4.49634928891205583165789611829, −4.11829857381933869486471570710, −2.49044428912180441855056970731, −1.56445103268616788442394224030, 0,
1.56445103268616788442394224030, 2.49044428912180441855056970731, 4.11829857381933869486471570710, 4.49634928891205583165789611829, 5.38874182297044395884066521337, 5.99319431655404024742384634043, 7.20498580264196947433326088967, 7.992316230922769723704960580517, 8.997401826351615974945902154762
