| L(s) = 1 | − 0.793·2-s + 2.98·3-s − 1.37·4-s − 2.40·5-s − 2.36·6-s − 3.11·7-s + 2.67·8-s + 5.91·9-s + 1.90·10-s − 11-s − 4.09·12-s + 3.11·13-s + 2.46·14-s − 7.18·15-s + 0.621·16-s + 3.72·17-s − 4.69·18-s − 5.31·19-s + 3.29·20-s − 9.29·21-s + 0.793·22-s + 1.33·23-s + 7.98·24-s + 0.787·25-s − 2.46·26-s + 8.69·27-s + 4.26·28-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + 1.72·3-s − 0.685·4-s − 1.07·5-s − 0.966·6-s − 1.17·7-s + 0.945·8-s + 1.97·9-s + 0.603·10-s − 0.301·11-s − 1.18·12-s + 0.863·13-s + 0.659·14-s − 1.85·15-s + 0.155·16-s + 0.902·17-s − 1.10·18-s − 1.21·19-s + 0.737·20-s − 2.02·21-s + 0.169·22-s + 0.277·23-s + 1.62·24-s + 0.157·25-s − 0.484·26-s + 1.67·27-s + 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 0.793T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.60T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 9.30T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.890316347717936996685139467235, −8.592147904272607412468271762823, −7.57188729513132714275372112795, −7.39247592594524286960178921268, −5.93374998519972111020008714439, −4.43695916415817499892835078056, −3.61931209912576462313940180655, −3.29401470946521362673829870656, −1.74723058653250238911709268798, 0,
1.74723058653250238911709268798, 3.29401470946521362673829870656, 3.61931209912576462313940180655, 4.43695916415817499892835078056, 5.93374998519972111020008714439, 7.39247592594524286960178921268, 7.57188729513132714275372112795, 8.592147904272607412468271762823, 8.890316347717936996685139467235
