| L(s) = 1 | − 4·2-s + 1.13·3-s + 16·4-s + 11.4·5-s − 4.52·6-s + 15.4·7-s − 64·8-s − 241.·9-s − 45.6·10-s + 596.·11-s + 18.0·12-s − 1.01e3·13-s − 61.9·14-s + 12.9·15-s + 256·16-s + 260.·17-s + 966.·18-s + 182.·20-s + 17.5·21-s − 2.38e3·22-s + 188.·23-s − 72.3·24-s − 2.99e3·25-s + 4.07e3·26-s − 548.·27-s + 247.·28-s + 2.99e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0725·3-s + 0.5·4-s + 0.204·5-s − 0.0513·6-s + 0.119·7-s − 0.353·8-s − 0.994·9-s − 0.144·10-s + 1.48·11-s + 0.0362·12-s − 1.67·13-s − 0.0844·14-s + 0.0148·15-s + 0.250·16-s + 0.218·17-s + 0.703·18-s + 0.102·20-s + 0.00866·21-s − 1.05·22-s + 0.0744·23-s − 0.0256·24-s − 0.958·25-s + 1.18·26-s − 0.144·27-s + 0.0597·28-s + 0.662·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.259260172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259260172\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.13T + 243T^{2} \) |
| 5 | \( 1 - 11.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 15.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.01e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 260.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 188.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.44e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.58e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−9.436935720936224164146628506682, −9.017136713752799902210016655609, −7.954217661836273955795430079282, −7.17045492817408737465434885922, −6.22686758794191592801105071214, −5.32491841692444091371159377472, −4.05218998036047267241456591381, −2.81085786411749520123139647950, −1.85886538696157878679773638084, −0.56881307686302487000255717933,
0.56881307686302487000255717933, 1.85886538696157878679773638084, 2.81085786411749520123139647950, 4.05218998036047267241456591381, 5.32491841692444091371159377472, 6.22686758794191592801105071214, 7.17045492817408737465434885922, 7.954217661836273955795430079282, 9.017136713752799902210016655609, 9.436935720936224164146628506682
