| L(s) = 1 | − 4.88·2-s − 8.09·4-s − 42.7·5-s + 196.·8-s + 209.·10-s − 710.·11-s − 885.·13-s − 699.·16-s − 701.·17-s + 1.25e3·19-s + 345.·20-s + 3.47e3·22-s + 1.04e3·23-s − 1.29e3·25-s + 4.33e3·26-s − 6.15e3·29-s − 2.29e3·31-s − 2.85e3·32-s + 3.42e3·34-s + 404.·37-s − 6.14e3·38-s − 8.38e3·40-s − 1.78e4·41-s − 1.46e4·43-s + 5.75e3·44-s − 5.11e3·46-s − 2.11e4·47-s + ⋯ |
| L(s) = 1 | − 0.864·2-s − 0.252·4-s − 0.764·5-s + 1.08·8-s + 0.661·10-s − 1.77·11-s − 1.45·13-s − 0.683·16-s − 0.588·17-s + 0.798·19-s + 0.193·20-s + 1.53·22-s + 0.412·23-s − 0.415·25-s + 1.25·26-s − 1.35·29-s − 0.428·31-s − 0.492·32-s + 0.508·34-s + 0.0485·37-s − 0.689·38-s − 0.828·40-s − 1.66·41-s − 1.20·43-s + 0.447·44-s − 0.356·46-s − 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.02808561305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02808561305\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.88T + 32T^{2} \) |
| 5 | \( 1 + 42.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 710.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 885.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 701.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 404.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.46e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 107.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.67e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.47e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.49e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e5T + 8.58e9T^{2} \) |
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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
−10.13164426774270196193304684425, −9.502709027721138181317802554519, −8.387838007238723146662700111521, −7.68905832038362787093906545176, −7.15621444491913960390251456913, −5.29943269706303214340559992123, −4.68607916599759017913458556336, −3.25990725986850750993033739667, −1.91431612029192322944467487077, −0.094662332482458449387151588158,
0.094662332482458449387151588158, 1.91431612029192322944467487077, 3.25990725986850750993033739667, 4.68607916599759017913458556336, 5.29943269706303214340559992123, 7.15621444491913960390251456913, 7.68905832038362787093906545176, 8.387838007238723146662700111521, 9.502709027721138181317802554519, 10.13164426774270196193304684425
