| L(s) = 1 | − 2.27·2-s − 2.82·4-s − 4.54·5-s + 24.6·8-s + 10.3·10-s + 40.7·11-s − 53.2·13-s − 33.4·16-s + 4.54·17-s − 122.·19-s + 12.8·20-s − 92.7·22-s − 131.·23-s − 104.·25-s + 121.·26-s + 216.·29-s + 251.·31-s − 120.·32-s − 10.3·34-s + 11.8·37-s + 278.·38-s − 112.·40-s − 111.·41-s + 369.·43-s − 115.·44-s + 298.·46-s − 262.·47-s + ⋯ |
| L(s) = 1 | − 0.804·2-s − 0.353·4-s − 0.406·5-s + 1.08·8-s + 0.327·10-s + 1.11·11-s − 1.13·13-s − 0.522·16-s + 0.0649·17-s − 1.48·19-s + 0.143·20-s − 0.898·22-s − 1.19·23-s − 0.834·25-s + 0.914·26-s + 1.38·29-s + 1.45·31-s − 0.668·32-s − 0.0522·34-s + 0.0528·37-s + 1.19·38-s − 0.442·40-s − 0.425·41-s + 1.30·43-s − 0.394·44-s + 0.957·46-s − 0.815·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7807484555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7807484555\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.27T + 8T^{2} \) |
| 5 | \( 1 + 4.54T + 125T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 637.T + 9.12e5T^{2} \) |
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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
−10.29312676949778676866035382490, −9.902982592825101480440539840035, −8.790806704786590083336837896943, −8.199470125953334476160793993620, −7.21673092787523816716364967762, −6.19164934299687274337274066824, −4.65342779442294263022442699013, −3.98311374031340365598012413412, −2.16558080544106776519264121809, −0.63506450749297929860961787280,
0.63506450749297929860961787280, 2.16558080544106776519264121809, 3.98311374031340365598012413412, 4.65342779442294263022442699013, 6.19164934299687274337274066824, 7.21673092787523816716364967762, 8.199470125953334476160793993620, 8.790806704786590083336837896943, 9.902982592825101480440539840035, 10.29312676949778676866035382490
