| L(s) = 1 | + 1.21·2-s + 3-s − 0.525·4-s + 1.21·6-s − 4.90·7-s − 3.06·8-s + 9-s − 11-s − 0.525·12-s − 4.14·13-s − 5.95·14-s − 2.67·16-s + 5.33·17-s + 1.21·18-s − 5.18·19-s − 4.90·21-s − 1.21·22-s − 4·23-s − 3.06·24-s − 5.03·26-s + 27-s + 2.57·28-s + 1.80·29-s + 2.62·31-s + 2.88·32-s − 33-s + 6.47·34-s + ⋯ |
| L(s) = 1 | + 0.858·2-s + 0.577·3-s − 0.262·4-s + 0.495·6-s − 1.85·7-s − 1.08·8-s + 0.333·9-s − 0.301·11-s − 0.151·12-s − 1.15·13-s − 1.59·14-s − 0.668·16-s + 1.29·17-s + 0.286·18-s − 1.18·19-s − 1.06·21-s − 0.258·22-s − 0.834·23-s − 0.625·24-s − 0.987·26-s + 0.192·27-s + 0.486·28-s + 0.335·29-s + 0.470·31-s + 0.510·32-s − 0.174·33-s + 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 0.428T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 - 0.622T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
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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
−9.989488157531560371503321545022, −9.060005868768446311906458682110, −8.162307526514416849187154306414, −7.00717280563667899036143555789, −6.21760680023177783022639434745, −5.30010138021793595739629337946, −4.15681673162548209544048057646, −3.32735084440142798165674678773, −2.56383985908226904712142055731, 0,
2.56383985908226904712142055731, 3.32735084440142798165674678773, 4.15681673162548209544048057646, 5.30010138021793595739629337946, 6.21760680023177783022639434745, 7.00717280563667899036143555789, 8.162307526514416849187154306414, 9.060005868768446311906458682110, 9.989488157531560371503321545022
