| L(s) = 1 | − 3.52·2-s − 19.6·4-s + 80.4·7-s + 181.·8-s − 520.·11-s − 421.·13-s − 283.·14-s − 12.3·16-s + 1.16e3·17-s + 1.17e3·19-s + 1.83e3·22-s + 2.37e3·23-s + 1.48e3·26-s − 1.57e3·28-s + 7.05e3·29-s − 5.56e3·31-s − 5.77e3·32-s − 4.09e3·34-s − 2.38e3·37-s − 4.12e3·38-s − 8.38e3·41-s − 2.05e4·43-s + 1.02e4·44-s − 8.36e3·46-s + 8.35e3·47-s − 1.03e4·49-s + 8.26e3·52-s + ⋯ |
| L(s) = 1 | − 0.622·2-s − 0.612·4-s + 0.620·7-s + 1.00·8-s − 1.29·11-s − 0.692·13-s − 0.386·14-s − 0.0120·16-s + 0.975·17-s + 0.743·19-s + 0.807·22-s + 0.935·23-s + 0.430·26-s − 0.380·28-s + 1.55·29-s − 1.03·31-s − 0.996·32-s − 0.607·34-s − 0.287·37-s − 0.463·38-s − 0.779·41-s − 1.69·43-s + 0.794·44-s − 0.582·46-s + 0.551·47-s − 0.615·49-s + 0.423·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.52T + 32T^{2} \) |
| 7 | \( 1 - 80.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.07e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.24e4T + 8.58e9T^{2} \) |
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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
−10.54138582761428206635334811570, −10.00263487673777013475350690389, −8.864863842424369576015871612326, −7.966548461927288362166301495835, −7.26521083420736437653496386604, −5.37321906525100004509934674565, −4.75259243107793026029078244010, −3.05523824434054866961339518125, −1.38038064097969684262948450517, 0,
1.38038064097969684262948450517, 3.05523824434054866961339518125, 4.75259243107793026029078244010, 5.37321906525100004509934674565, 7.26521083420736437653496386604, 7.966548461927288362166301495835, 8.864863842424369576015871612326, 10.00263487673777013475350690389, 10.54138582761428206635334811570
