| L(s) = 1 | − 9.15·2-s + 9·3-s + 51.8·4-s − 82.3·6-s + 226.·7-s − 181.·8-s + 81·9-s + 121·11-s + 466.·12-s − 60.5·13-s − 2.07e3·14-s + 2.66·16-s + 1.98e3·17-s − 741.·18-s − 1.09e3·19-s + 2.03e3·21-s − 1.10e3·22-s − 4.63e3·23-s − 1.63e3·24-s + 553.·26-s + 729·27-s + 1.17e4·28-s + 8.56e3·29-s − 409.·31-s + 5.78e3·32-s + 1.08e3·33-s − 1.81e4·34-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.61·4-s − 0.934·6-s + 1.74·7-s − 1.00·8-s + 0.333·9-s + 0.301·11-s + 0.934·12-s − 0.0992·13-s − 2.82·14-s + 0.00260·16-s + 1.66·17-s − 0.539·18-s − 0.695·19-s + 1.00·21-s − 0.487·22-s − 1.82·23-s − 0.578·24-s + 0.160·26-s + 0.192·27-s + 2.83·28-s + 1.89·29-s − 0.0765·31-s + 0.997·32-s + 0.174·33-s − 2.69·34-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
| \(L(3)\) |
\(\approx\) |
\(1.791511437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.791511437\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.15T + 32T^{2} \) |
| 7 | \( 1 - 226.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 60.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.09e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 409.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.43e4T + 8.58e9T^{2} \) |
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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
−9.340640959434214150138760157334, −8.453894332176544281238572935721, −8.030505760539681625177945774618, −7.50707936238239675709523707141, −6.33483149040921055622752983883, −5.04460026658464617876631209646, −3.95666772852819011636980622781, −2.40991948531466121625052493945, −1.62100240914570564198536896943, −0.813300461132641414589890166832,
0.813300461132641414589890166832, 1.62100240914570564198536896943, 2.40991948531466121625052493945, 3.95666772852819011636980622781, 5.04460026658464617876631209646, 6.33483149040921055622752983883, 7.50707936238239675709523707141, 8.030505760539681625177945774618, 8.453894332176544281238572935721, 9.340640959434214150138760157334
