| L(s) = 1 | − 0.790·2-s + 3·3-s − 7.37·4-s + 7.57·5-s − 2.37·6-s + 7·7-s + 12.1·8-s + 9·9-s − 5.98·10-s + 30.8·11-s − 22.1·12-s − 34.0·13-s − 5.53·14-s + 22.7·15-s + 49.3·16-s − 67.3·17-s − 7.11·18-s + 140.·19-s − 55.8·20-s + 21·21-s − 24.4·22-s − 23·23-s + 36.4·24-s − 67.6·25-s + 26.9·26-s + 27·27-s − 51.6·28-s + ⋯ |
| L(s) = 1 | − 0.279·2-s + 0.577·3-s − 0.921·4-s + 0.677·5-s − 0.161·6-s + 0.377·7-s + 0.537·8-s + 0.333·9-s − 0.189·10-s + 0.846·11-s − 0.532·12-s − 0.726·13-s − 0.105·14-s + 0.391·15-s + 0.771·16-s − 0.960·17-s − 0.0931·18-s + 1.69·19-s − 0.624·20-s + 0.218·21-s − 0.236·22-s − 0.208·23-s + 0.310·24-s − 0.540·25-s + 0.203·26-s + 0.192·27-s − 0.348·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.066955908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066955908\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 0.790T + 8T^{2} \) |
| 5 | \( 1 - 7.57T + 125T^{2} \) |
| 11 | \( 1 - 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 59.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 49.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 118.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 405.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 280.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 551.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 178.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 662.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 498.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 971.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 101.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 289.T + 9.12e5T^{2} \) |
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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
−10.17748499316640662283666620010, −9.514712892250407243009879404797, −9.006237535509485724633770400549, −7.991721630708203254924752731941, −7.12781121053811998521820628008, −5.77755160569957319258906065313, −4.75502079173009321421741394248, −3.78364398783060202152900824363, −2.28448754881082239011627858039, −0.972526635266434563349657489172,
0.972526635266434563349657489172, 2.28448754881082239011627858039, 3.78364398783060202152900824363, 4.75502079173009321421741394248, 5.77755160569957319258906065313, 7.12781121053811998521820628008, 7.991721630708203254924752731941, 9.006237535509485724633770400549, 9.514712892250407243009879404797, 10.17748499316640662283666620010
