| L(s) = 1 | + 1.63·2-s + 0.767·3-s + 0.665·4-s + 1.25·6-s − 1.80·7-s − 2.17·8-s − 2.41·9-s − 0.738·11-s + 0.511·12-s + 6.48·13-s − 2.95·14-s − 4.88·16-s + 3.64·17-s − 3.93·18-s − 1.38·21-s − 1.20·22-s − 1.77·23-s − 1.67·24-s + 10.5·26-s − 4.15·27-s − 1.20·28-s − 3.42·29-s − 1.39·31-s − 3.62·32-s − 0.566·33-s + 5.95·34-s − 1.60·36-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.443·3-s + 0.332·4-s + 0.511·6-s − 0.682·7-s − 0.770·8-s − 0.803·9-s − 0.222·11-s + 0.147·12-s + 1.79·13-s − 0.788·14-s − 1.22·16-s + 0.884·17-s − 0.927·18-s − 0.302·21-s − 0.256·22-s − 0.369·23-s − 0.341·24-s + 2.07·26-s − 0.799·27-s − 0.227·28-s − 0.635·29-s − 0.251·31-s − 0.640·32-s − 0.0986·33-s + 1.02·34-s − 0.267·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 3 | \( 1 - 0.767T + 3T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 0.738T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 + 0.383T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 0.631T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.33831064066335547686791753561, −6.33014258740802738182616922281, −5.79790986194224435752355622009, −5.61312261841789717020174516494, −4.34707205245143270559342073050, −3.87540034828543430620630843069, −3.08997940176114018110158760399, −2.76778269482592443923952706828, −1.40475007442179336201539067463, 0,
1.40475007442179336201539067463, 2.76778269482592443923952706828, 3.08997940176114018110158760399, 3.87540034828543430620630843069, 4.34707205245143270559342073050, 5.61312261841789717020174516494, 5.79790986194224435752355622009, 6.33014258740802738182616922281, 7.33831064066335547686791753561
