L(s) = 1 | − 43.0·2-s + 78.8·3-s − 194.·4-s − 1.12e4·5-s − 3.39e3·6-s − 9.28e3·7-s + 9.65e4·8-s − 1.70e5·9-s + 4.85e5·10-s − 7.75e5·11-s − 1.53e4·12-s − 1.35e6·13-s + 3.99e5·14-s − 8.89e5·15-s − 3.75e6·16-s + 5.04e6·17-s + 7.35e6·18-s + 1.10e7·19-s + 2.18e6·20-s − 7.32e5·21-s + 3.34e7·22-s − 6.43e6·23-s + 7.61e6·24-s + 7.83e7·25-s + 5.84e7·26-s − 2.74e7·27-s + 1.80e6·28-s + ⋯ |
L(s) = 1 | − 0.951·2-s + 0.187·3-s − 0.0947·4-s − 1.61·5-s − 0.178·6-s − 0.208·7-s + 1.04·8-s − 0.964·9-s + 1.53·10-s − 1.45·11-s − 0.0177·12-s − 1.01·13-s + 0.198·14-s − 0.302·15-s − 0.896·16-s + 0.862·17-s + 0.918·18-s + 1.02·19-s + 0.152·20-s − 0.0391·21-s + 1.38·22-s − 0.208·23-s + 0.195·24-s + 1.60·25-s + 0.964·26-s − 0.368·27-s + 0.0197·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.2954160734\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2954160734\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + 6.43e6T \) |
good | 2 | \( 1 + 43.0T + 2.04e3T^{2} \) |
| 3 | \( 1 - 78.8T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.12e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 9.28e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.75e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.35e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.04e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.10e7T + 1.16e14T^{2} \) |
| 29 | \( 1 + 3.22e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.54e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.12e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.65e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.75e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.34e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.77e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.57e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.63e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.06e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.06e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.53e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.42e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.34e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.54e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.36e10T + 7.15e21T^{2} \) |
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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
−15.55334368322934735163214931894, −14.10759521342452228086202750440, −12.41388508406357045337550549372, −11.12038943879527006722490321698, −9.724102876394529503030005242495, −8.111210586299896866854035188583, −7.65912790135869598503074910408, −4.91638655469529011677625270946, −3.06737697485531784348868406772, −0.42104002175365072270312971029,
0.42104002175365072270312971029, 3.06737697485531784348868406772, 4.91638655469529011677625270946, 7.65912790135869598503074910408, 8.111210586299896866854035188583, 9.724102876394529503030005242495, 11.12038943879527006722490321698, 12.41388508406357045337550549372, 14.10759521342452228086202750440, 15.55334368322934735163214931894