| L(s) = 1 | − 2.31·3-s + 0.313·5-s + 2.05·7-s + 2.34·9-s − 1.08·11-s + 2.96·13-s − 0.724·15-s + 5.04·17-s + 1.75·19-s − 4.75·21-s − 5.45·23-s − 4.90·25-s + 1.51·27-s − 4.82·29-s − 3.99·31-s + 2.50·33-s + 0.645·35-s + 11.2·37-s − 6.84·39-s − 8.82·41-s − 5.01·43-s + 0.735·45-s + 4.60·47-s − 2.76·49-s − 11.6·51-s − 7.97·53-s − 0.340·55-s + ⋯ |
| L(s) = 1 | − 1.33·3-s + 0.140·5-s + 0.777·7-s + 0.781·9-s − 0.327·11-s + 0.821·13-s − 0.187·15-s + 1.22·17-s + 0.403·19-s − 1.03·21-s − 1.13·23-s − 0.980·25-s + 0.291·27-s − 0.895·29-s − 0.717·31-s + 0.436·33-s + 0.109·35-s + 1.85·37-s − 1.09·39-s − 1.37·41-s − 0.764·43-s + 0.109·45-s + 0.672·47-s − 0.395·49-s − 1.63·51-s − 1.09·53-s − 0.0458·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 - 0.313T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 7.97T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 + 0.0874T + 73T^{2} \) |
| 79 | \( 1 - 6.33T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−7.62386203535966500892809034603, −6.58992849250537022482828977530, −5.84963128648381281735465250385, −5.60164638248484032890033316475, −4.86242273216525476896444515434, −4.05348863117404591141024674104, −3.22844109951486051213431662298, −1.92367986882436512274049697936, −1.18142570961746241554222612551, 0,
1.18142570961746241554222612551, 1.92367986882436512274049697936, 3.22844109951486051213431662298, 4.05348863117404591141024674104, 4.86242273216525476896444515434, 5.60164638248484032890033316475, 5.84963128648381281735465250385, 6.58992849250537022482828977530, 7.62386203535966500892809034603
