| L(s) = 1 | − 1.53·2-s + 3-s + 0.369·4-s − 1.53·6-s − 0.290·7-s + 2.51·8-s + 9-s − 11-s + 0.369·12-s − 6.97·13-s + 0.447·14-s − 4.60·16-s − 4.78·17-s − 1.53·18-s + 7.75·19-s − 0.290·21-s + 1.53·22-s − 4·23-s + 2.51·24-s + 10.7·26-s + 27-s − 0.107·28-s − 7.41·29-s + 6.34·31-s + 2.06·32-s − 33-s + 7.36·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.184·4-s − 0.628·6-s − 0.109·7-s + 0.887·8-s + 0.333·9-s − 0.301·11-s + 0.106·12-s − 1.93·13-s + 0.119·14-s − 1.15·16-s − 1.16·17-s − 0.362·18-s + 1.77·19-s − 0.0634·21-s + 0.328·22-s − 0.834·23-s + 0.512·24-s + 2.10·26-s + 0.192·27-s − 0.0202·28-s − 1.37·29-s + 1.13·31-s + 0.364·32-s − 0.174·33-s + 1.26·34-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 | \( 1 + 1.53T + 2T^{2} \) |
| 7 | \( 1 + 0.290T + 7T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.290T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 + 5.07T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 0.921T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−9.694158568140671318089644424845, −9.153010968813751776226261488951, −8.052947122976830871173468382930, −7.58641999983254212492816478410, −6.76909145900880626992601319300, −5.20577822701259887684677418348, −4.39573475830108186797434377355, −2.93043922315751549885649373895, −1.80804594194448207286975864668, 0,
1.80804594194448207286975864668, 2.93043922315751549885649373895, 4.39573475830108186797434377355, 5.20577822701259887684677418348, 6.76909145900880626992601319300, 7.58641999983254212492816478410, 8.052947122976830871173468382930, 9.153010968813751776226261488951, 9.694158568140671318089644424845
