| L(s) = 1 | − 3-s + 2.48·5-s − 0.0448·7-s + 9-s − 0.683·11-s − 6.72·13-s − 2.48·15-s − 1.19·17-s + 1.53·19-s + 0.0448·21-s + 23-s + 1.15·25-s − 27-s − 29-s + 1.40·31-s + 0.683·33-s − 0.111·35-s + 8.68·37-s + 6.72·39-s + 11.5·41-s − 6.28·43-s + 2.48·45-s + 5.00·47-s − 6.99·49-s + 1.19·51-s − 1.85·53-s − 1.69·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.10·5-s − 0.0169·7-s + 0.333·9-s − 0.206·11-s − 1.86·13-s − 0.640·15-s − 0.289·17-s + 0.351·19-s + 0.00978·21-s + 0.208·23-s + 0.231·25-s − 0.192·27-s − 0.185·29-s + 0.252·31-s + 0.118·33-s − 0.0188·35-s + 1.42·37-s + 1.07·39-s + 1.80·41-s − 0.958·43-s + 0.369·45-s + 0.729·47-s − 0.999·49-s + 0.167·51-s − 0.254·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 + 0.0448T + 7T^{2} \) |
| 11 | \( 1 + 0.683T + 11T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 + 0.0568T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 0.0825T + 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + 0.473T + 89T^{2} \) |
| 97 | \( 1 + 5.20T + 97T^{2} \) |
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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
−7.48474226584635283252094980760, −6.66922460352753795612891370882, −6.09139946335339628711134058798, −5.36545967214146839660792552090, −4.88016498295292940436411707479, −4.11531450904470545288277800006, −2.78646473486968306180134515911, −2.31549568410892928583216848976, −1.26991776593545882670296912952, 0,
1.26991776593545882670296912952, 2.31549568410892928583216848976, 2.78646473486968306180134515911, 4.11531450904470545288277800006, 4.88016498295292940436411707479, 5.36545967214146839660792552090, 6.09139946335339628711134058798, 6.66922460352753795612891370882, 7.48474226584635283252094980760
