| L(s) = 1 | + 2.32·2-s + 3·3-s − 2.57·4-s + 6.98·6-s − 22.4·7-s − 24.6·8-s + 9·9-s + 11·11-s − 7.72·12-s + 9.86·13-s − 52.3·14-s − 36.7·16-s + 128.·17-s + 20.9·18-s + 7.04·19-s − 67.4·21-s + 25.6·22-s − 0.654·23-s − 73.8·24-s + 22.9·26-s + 27·27-s + 57.8·28-s − 229.·29-s + 155.·31-s + 111.·32-s + 33·33-s + 298.·34-s + ⋯ |
| L(s) = 1 | + 0.823·2-s + 0.577·3-s − 0.321·4-s + 0.475·6-s − 1.21·7-s − 1.08·8-s + 0.333·9-s + 0.301·11-s − 0.185·12-s + 0.210·13-s − 0.998·14-s − 0.574·16-s + 1.82·17-s + 0.274·18-s + 0.0850·19-s − 0.700·21-s + 0.248·22-s − 0.00593·23-s − 0.628·24-s + 0.173·26-s + 0.192·27-s + 0.390·28-s − 1.46·29-s + 0.902·31-s + 0.615·32-s + 0.174·33-s + 1.50·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.840764868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.840764868\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.32T + 8T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 13 | \( 1 - 9.86T + 2.19e3T^{2} \) |
| 17 | \( 1 - 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.04T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.654T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 651.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 423.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 374.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 96.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 24.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 730.T + 9.12e5T^{2} \) |
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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
−9.566689025189692554239618416207, −9.260930493310804038584174733051, −8.141253089791682904017279604648, −7.19538017647700608216589101276, −6.08708190831412417499210704726, −5.48565476795241888890576898314, −4.12390094827828533918595865420, −3.51096882456687860401636107810, −2.66408071374131149618080691913, −0.812636248968871468148556731499,
0.812636248968871468148556731499, 2.66408071374131149618080691913, 3.51096882456687860401636107810, 4.12390094827828533918595865420, 5.48565476795241888890576898314, 6.08708190831412417499210704726, 7.19538017647700608216589101276, 8.141253089791682904017279604648, 9.260930493310804038584174733051, 9.566689025189692554239618416207
