| L(s) = 1 | + 0.755·3-s + 4.26·5-s − 2.09·7-s − 2.42·9-s − 4.58·11-s + 1.90·13-s + 3.22·15-s − 5.40·17-s − 19-s − 1.58·21-s − 4.46·23-s + 13.2·25-s − 4.10·27-s + 1.04·29-s − 6.77·31-s − 3.46·33-s − 8.92·35-s − 1.41·37-s + 1.44·39-s + 4.93·41-s + 1.29·43-s − 10.3·45-s + 8.54·47-s − 2.62·49-s − 4.08·51-s + 53-s − 19.5·55-s + ⋯ |
| L(s) = 1 | + 0.436·3-s + 1.90·5-s − 0.790·7-s − 0.809·9-s − 1.38·11-s + 0.528·13-s + 0.833·15-s − 1.31·17-s − 0.229·19-s − 0.344·21-s − 0.931·23-s + 2.64·25-s − 0.789·27-s + 0.193·29-s − 1.21·31-s − 0.602·33-s − 1.50·35-s − 0.232·37-s + 0.230·39-s + 0.770·41-s + 0.197·43-s − 1.54·45-s + 1.24·47-s − 0.375·49-s − 0.572·51-s + 0.137·53-s − 2.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 0.755T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 + 2.09T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.99T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 4.71T + 71T^{2} \) |
| 73 | \( 1 + 6.64T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−8.290744080979944379325901619349, −7.27778846444547897677390672125, −6.39653250608277548350511350718, −5.83130974702548721042896159616, −5.41192018481290903983214050277, −4.27983585158325744533311283260, −3.00830826213383015103800052942, −2.51842149490043962200220957807, −1.77232324285410911951801193045, 0,
1.77232324285410911951801193045, 2.51842149490043962200220957807, 3.00830826213383015103800052942, 4.27983585158325744533311283260, 5.41192018481290903983214050277, 5.83130974702548721042896159616, 6.39653250608277548350511350718, 7.27778846444547897677390672125, 8.290744080979944379325901619349
