| L(s) = 1 | − 6.66·3-s − 25·5-s + 12.8·7-s − 198.·9-s + 121·11-s − 485.·13-s + 166.·15-s + 266.·17-s + 149.·19-s − 85.8·21-s + 3.21e3·23-s + 625·25-s + 2.94e3·27-s + 2.94e3·29-s − 2.14e3·31-s − 806.·33-s − 322.·35-s − 808.·37-s + 3.23e3·39-s + 1.01e4·41-s − 2.76e3·43-s + 4.96e3·45-s − 9.97e3·47-s − 1.66e4·49-s − 1.77e3·51-s + 7.12e3·53-s − 3.02e3·55-s + ⋯ |
| L(s) = 1 | − 0.427·3-s − 0.447·5-s + 0.0993·7-s − 0.817·9-s + 0.301·11-s − 0.796·13-s + 0.191·15-s + 0.223·17-s + 0.0951·19-s − 0.0424·21-s + 1.26·23-s + 0.200·25-s + 0.776·27-s + 0.651·29-s − 0.401·31-s − 0.128·33-s − 0.0444·35-s − 0.0970·37-s + 0.340·39-s + 0.938·41-s − 0.227·43-s + 0.365·45-s − 0.658·47-s − 0.990·49-s − 0.0956·51-s + 0.348·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 + 6.66T + 243T^{2} \) |
| 7 | \( 1 - 12.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 266.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 808.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.50e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−8.946418829708460948678659560340, −8.141793435404070887678580272096, −7.22824628762037884995194921858, −6.41327372352658090451176896243, −5.36460903244616505964106359499, −4.68507709564285667517265114850, −3.45422268186525848691085801530, −2.51809885086900133611691236415, −1.04191821947363847817112434754, 0,
1.04191821947363847817112434754, 2.51809885086900133611691236415, 3.45422268186525848691085801530, 4.68507709564285667517265114850, 5.36460903244616505964106359499, 6.41327372352658090451176896243, 7.22824628762037884995194921858, 8.141793435404070887678580272096, 8.946418829708460948678659560340
