| L(s) = 1 | − 3.30·3-s + 0.595·5-s − 0.878·7-s + 7.92·9-s + 1.07·11-s + 3.70·13-s − 1.96·15-s − 5.87·17-s − 2.12·19-s + 2.90·21-s + 1.93·23-s − 4.64·25-s − 16.2·27-s − 9.16·29-s + 7.16·31-s − 3.54·33-s − 0.523·35-s + 6.81·37-s − 12.2·39-s + 3.27·41-s − 11.6·43-s + 4.72·45-s + 13.2·47-s − 6.22·49-s + 19.4·51-s + 11.0·53-s + 0.639·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.266·5-s − 0.331·7-s + 2.64·9-s + 0.323·11-s + 1.02·13-s − 0.508·15-s − 1.42·17-s − 0.488·19-s + 0.633·21-s + 0.403·23-s − 0.928·25-s − 3.13·27-s − 1.70·29-s + 1.28·31-s − 0.617·33-s − 0.0884·35-s + 1.11·37-s − 1.96·39-s + 0.510·41-s − 1.77·43-s + 0.704·45-s + 1.93·47-s − 0.889·49-s + 2.71·51-s + 1.52·53-s + 0.0861·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.595T + 5T^{2} \) |
| 7 | \( 1 + 0.878T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 - 7.16T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 - 9.75T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.930582495152537604955551123339, −6.97525201099267597287048184530, −6.40302637183620703110655173376, −6.00227048957491975019433755728, −5.24812530487756698910054667134, −4.35291303005970721894188762396, −3.81543050120512934476831834757, −2.18222004965771762916260299711, −1.13785571757702803400160570813, 0,
1.13785571757702803400160570813, 2.18222004965771762916260299711, 3.81543050120512934476831834757, 4.35291303005970721894188762396, 5.24812530487756698910054667134, 6.00227048957491975019433755728, 6.40302637183620703110655173376, 6.97525201099267597287048184530, 7.930582495152537604955551123339
