| L(s) = 1 | − 3.30·3-s + 2.30·5-s − 2.60·7-s + 7.90·9-s + 2.30·11-s − 1.30·13-s − 7.60·15-s − 6·17-s − 2·19-s + 8.60·21-s + 3.90·23-s + 0.302·25-s − 16.2·27-s + 3.90·29-s − 0.302·31-s − 7.60·33-s − 6·35-s − 37-s + 4.30·39-s + 9.90·41-s − 0.605·43-s + 18.2·45-s + 4.60·47-s − 0.211·49-s + 19.8·51-s + 6·53-s + 5.30·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 1.02·5-s − 0.984·7-s + 2.63·9-s + 0.694·11-s − 0.361·13-s − 1.96·15-s − 1.45·17-s − 0.458·19-s + 1.87·21-s + 0.814·23-s + 0.0605·25-s − 3.11·27-s + 0.725·29-s − 0.0543·31-s − 1.32·33-s − 1.01·35-s − 0.164·37-s + 0.688·39-s + 1.54·41-s − 0.0923·43-s + 2.71·45-s + 0.671·47-s − 0.0301·49-s + 2.77·51-s + 0.824·53-s + 0.715·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.985017198244569993205861149383, −7.39662100747186063632623998386, −6.55753432047408324251295934864, −6.37134261285009509756808409962, −5.63076875998154267247011087921, −4.76280545299010122828466146824, −4.03968676816200616349597357438, −2.48542401981367028793302196799, −1.28093493432064392838855926278, 0,
1.28093493432064392838855926278, 2.48542401981367028793302196799, 4.03968676816200616349597357438, 4.76280545299010122828466146824, 5.63076875998154267247011087921, 6.37134261285009509756808409962, 6.55753432047408324251295934864, 7.39662100747186063632623998386, 8.985017198244569993205861149383
