| L(s) = 1 | + 9·3-s − 46.1·5-s + 59.9·7-s + 81·9-s + 400.·11-s − 351.·13-s − 415.·15-s + 1.66e3·17-s + 564.·19-s + 539.·21-s − 3.83e3·23-s − 994.·25-s + 729·27-s + 5.94e3·29-s + 2.67e3·31-s + 3.60e3·33-s − 2.76e3·35-s − 771.·37-s − 3.16e3·39-s + 5.11e3·41-s − 9.51e3·43-s − 3.73e3·45-s − 8.13e3·47-s − 1.32e4·49-s + 1.49e4·51-s − 1.10e4·53-s − 1.84e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.825·5-s + 0.462·7-s + 0.333·9-s + 0.996·11-s − 0.577·13-s − 0.476·15-s + 1.39·17-s + 0.358·19-s + 0.266·21-s − 1.51·23-s − 0.318·25-s + 0.192·27-s + 1.31·29-s + 0.500·31-s + 0.575·33-s − 0.381·35-s − 0.0926·37-s − 0.333·39-s + 0.475·41-s − 0.785·43-s − 0.275·45-s − 0.537·47-s − 0.786·49-s + 0.805·51-s − 0.540·53-s − 0.823·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.687391636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687391636\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| good | 5 | \( 1 + 46.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 59.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 400.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 564.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 771.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.51e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.23e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.58e5T + 8.58e9T^{2} \) |
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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
−9.695987990752814043425832893869, −8.447701619539342228957415863285, −7.990562540987930008867250769063, −7.19066712930762135534513912661, −6.13030212632280204823464827997, −4.88961607805879797419179492134, −3.98787599612290740667420887786, −3.19960495410974915025253653855, −1.87874411251535549999999289861, −0.75332784551877371813971327839,
0.75332784551877371813971327839, 1.87874411251535549999999289861, 3.19960495410974915025253653855, 3.98787599612290740667420887786, 4.88961607805879797419179492134, 6.13030212632280204823464827997, 7.19066712930762135534513912661, 7.990562540987930008867250769063, 8.447701619539342228957415863285, 9.695987990752814043425832893869
