| L(s) = 1 | + 0.180·2-s − 1.96·4-s − 4.41·7-s − 0.714·8-s − 4.27·11-s + 2.79·13-s − 0.795·14-s + 3.80·16-s − 4.43·17-s + 2.43·19-s − 0.769·22-s + 5.77·23-s + 0.504·26-s + 8.68·28-s − 0.409·29-s + 7.79·31-s + 2.11·32-s − 0.799·34-s + 37-s + 0.438·38-s − 0.757·41-s − 2.19·43-s + 8.40·44-s + 1.04·46-s − 4.26·47-s + 12.4·49-s − 5.50·52-s + ⋯ |
| L(s) = 1 | + 0.127·2-s − 0.983·4-s − 1.66·7-s − 0.252·8-s − 1.28·11-s + 0.776·13-s − 0.212·14-s + 0.951·16-s − 1.07·17-s + 0.558·19-s − 0.164·22-s + 1.20·23-s + 0.0989·26-s + 1.64·28-s − 0.0759·29-s + 1.39·31-s + 0.373·32-s − 0.137·34-s + 0.164·37-s + 0.0711·38-s − 0.118·41-s − 0.335·43-s + 1.26·44-s + 0.153·46-s − 0.622·47-s + 1.78·49-s − 0.763·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.180T + 2T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 0.409T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 41 | \( 1 + 0.757T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 + 0.137T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.20T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.43799378665084992263797582530, −6.62880962977020729704977835060, −6.12415655832762544365240815574, −5.28505435280743280970813350242, −4.72235150457649383808432815713, −3.78577952998155199829324807114, −3.16898495340372337047632631716, −2.53024373215375016391146005964, −0.916527375267551843434235415542, 0,
0.916527375267551843434235415542, 2.53024373215375016391146005964, 3.16898495340372337047632631716, 3.78577952998155199829324807114, 4.72235150457649383808432815713, 5.28505435280743280970813350242, 6.12415655832762544365240815574, 6.62880962977020729704977835060, 7.43799378665084992263797582530
