| L(s) = 1 | + 7.81·2-s − 9·3-s + 29.0·4-s − 70.3·6-s − 11.0·7-s − 22.8·8-s + 81·9-s + 121·11-s − 261.·12-s − 248.·13-s − 86.2·14-s − 1.10e3·16-s + 669.·17-s + 633.·18-s + 2.32e3·19-s + 99.2·21-s + 945.·22-s − 867.·23-s + 205.·24-s − 1.94e3·26-s − 729·27-s − 320.·28-s + 2.64e3·29-s − 3.38e3·31-s − 7.93e3·32-s − 1.08e3·33-s + 5.23e3·34-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 0.577·3-s + 0.908·4-s − 0.797·6-s − 0.0851·7-s − 0.126·8-s + 0.333·9-s + 0.301·11-s − 0.524·12-s − 0.408·13-s − 0.117·14-s − 1.08·16-s + 0.561·17-s + 0.460·18-s + 1.47·19-s + 0.0491·21-s + 0.416·22-s − 0.342·23-s + 0.0729·24-s − 0.564·26-s − 0.192·27-s − 0.0773·28-s + 0.583·29-s − 0.631·31-s − 1.36·32-s − 0.174·33-s + 0.775·34-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 | \( 1 - 7.81T + 32T^{2} \) |
| 7 | \( 1 + 11.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 248.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 669.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 867.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 434.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.36e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.94e4T + 8.58e9T^{2} \) |
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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
−9.270001100627602779350356565233, −7.937999523135582719739067383846, −6.98319544352021917931994790192, −6.16128074517524313915689144401, −5.36016915994344026622296451659, −4.71678804437695634241431431638, −3.68062139497249392594100069469, −2.85553047917601724265965841289, −1.41702323250103529787098173275, 0,
1.41702323250103529787098173275, 2.85553047917601724265965841289, 3.68062139497249392594100069469, 4.71678804437695634241431431638, 5.36016915994344026622296451659, 6.16128074517524313915689144401, 6.98319544352021917931994790192, 7.937999523135582719739067383846, 9.270001100627602779350356565233
