| L(s) = 1 | + 4.16·2-s − 3.29·3-s + 9.34·4-s − 10.5·5-s − 13.7·6-s + 5.61·8-s − 16.1·9-s − 44.0·10-s − 54.8·11-s − 30.7·12-s − 30.0·13-s + 34.8·15-s − 51.3·16-s − 44.7·17-s − 67.2·18-s + 71.5·19-s − 98.8·20-s − 228.·22-s − 167.·23-s − 18.5·24-s − 13.2·25-s − 125.·26-s + 142.·27-s + 18.8·29-s + 145.·30-s − 114.·31-s − 258.·32-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.633·3-s + 1.16·4-s − 0.945·5-s − 0.933·6-s + 0.248·8-s − 0.598·9-s − 1.39·10-s − 1.50·11-s − 0.740·12-s − 0.640·13-s + 0.599·15-s − 0.802·16-s − 0.638·17-s − 0.880·18-s + 0.864·19-s − 1.10·20-s − 2.21·22-s − 1.51·23-s − 0.157·24-s − 0.106·25-s − 0.943·26-s + 1.01·27-s + 0.120·29-s + 0.882·30-s − 0.662·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7004590344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7004590344\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.16T + 8T^{2} \) |
| 3 | \( 1 + 3.29T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 + 54.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 541.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 486.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 703.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 352.T + 9.12e5T^{2} \) |
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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
−8.361159967089034875319021366331, −7.61895749900089309366976577148, −6.94905873517783356369305612275, −5.74445460974792807070162263620, −5.56565983889943575546678946222, −4.63196738835816796865425902387, −3.98943028408801681490059043272, −2.99174467875930142095093660047, −2.29945455283182407740978186299, −0.28023187976460659307436911197,
0.28023187976460659307436911197, 2.29945455283182407740978186299, 2.99174467875930142095093660047, 3.98943028408801681490059043272, 4.63196738835816796865425902387, 5.56565983889943575546678946222, 5.74445460974792807070162263620, 6.94905873517783356369305612275, 7.61895749900089309366976577148, 8.361159967089034875319021366331
