| L(s) = 1 | + 1.30·3-s − 2.99·5-s − 4.27·7-s − 1.30·9-s − 0.296·11-s + 3.69·13-s − 3.89·15-s − 0.405·17-s + 2.26·19-s − 5.56·21-s + 8.27·23-s + 3.94·25-s − 5.60·27-s + 5.34·29-s + 1.53·31-s − 0.386·33-s + 12.8·35-s + 1.82·37-s + 4.81·39-s − 3.79·41-s − 3.90·43-s + 3.90·45-s + 7.57·47-s + 11.3·49-s − 0.527·51-s + 1.82·53-s + 0.887·55-s + ⋯ |
| L(s) = 1 | + 0.751·3-s − 1.33·5-s − 1.61·7-s − 0.435·9-s − 0.0894·11-s + 1.02·13-s − 1.00·15-s − 0.0983·17-s + 0.520·19-s − 1.21·21-s + 1.72·23-s + 0.789·25-s − 1.07·27-s + 0.993·29-s + 0.275·31-s − 0.0672·33-s + 2.16·35-s + 0.299·37-s + 0.770·39-s − 0.593·41-s − 0.595·43-s + 0.582·45-s + 1.10·47-s + 1.61·49-s − 0.0739·51-s + 0.250·53-s + 0.119·55-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 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 + 0.296T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 + 0.405T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 - 7.57T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.61T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 + 2.83T + 97T^{2} \) |
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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
−7.47337187246928252604776907843, −6.92305221923331963683102953684, −6.25340554311286846612017060539, −5.44140164018326362940618426909, −4.39137371426424008301809294746, −3.64950562985144793036501563222, −3.14292873049017210347192042328, −2.71602537237286818361340080552, −1.06241324612747754033455873089, 0,
1.06241324612747754033455873089, 2.71602537237286818361340080552, 3.14292873049017210347192042328, 3.64950562985144793036501563222, 4.39137371426424008301809294746, 5.44140164018326362940618426909, 6.25340554311286846612017060539, 6.92305221923331963683102953684, 7.47337187246928252604776907843
