| L(s) = 1 | + 2.37·2-s − 3-s + 3.66·4-s − 2.37·6-s − 7-s + 3.95·8-s + 9-s + 1.57·11-s − 3.66·12-s − 13-s − 2.37·14-s + 2.08·16-s − 4.75·17-s + 2.37·18-s − 2.23·19-s + 21-s + 3.74·22-s − 5.84·23-s − 3.95·24-s − 2.37·26-s − 27-s − 3.66·28-s + 4.23·29-s + 7.28·31-s − 2.94·32-s − 1.57·33-s − 11.3·34-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.971·6-s − 0.377·7-s + 1.39·8-s + 0.333·9-s + 0.475·11-s − 1.05·12-s − 0.277·13-s − 0.635·14-s + 0.521·16-s − 1.15·17-s + 0.560·18-s − 0.513·19-s + 0.218·21-s + 0.799·22-s − 1.21·23-s − 0.807·24-s − 0.466·26-s − 0.192·27-s − 0.692·28-s + 0.786·29-s + 1.30·31-s − 0.520·32-s − 0.274·33-s − 1.94·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 + 0.913T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.09082395398105920497794303480, −6.62135180711349322856462594509, −6.15618213201619403004329517371, −5.45419471882198326642552180419, −4.57560709740909190022191259761, −4.27819732416698600139942581816, −3.38799020803252584301652601120, −2.53732933501111438252453360909, −1.66967461128626140280419849002, 0,
1.66967461128626140280419849002, 2.53732933501111438252453360909, 3.38799020803252584301652601120, 4.27819732416698600139942581816, 4.57560709740909190022191259761, 5.45419471882198326642552180419, 6.15618213201619403004329517371, 6.62135180711349322856462594509, 7.09082395398105920497794303480
