| L(s) = 1 | − 2.44·5-s + 2.44·7-s − 2·11-s − 3.46·13-s + 2.82·17-s − 2.82·19-s + 6.92·23-s + 0.999·25-s − 2.44·29-s + 7.34·31-s − 5.99·35-s + 10.3·37-s − 8.48·41-s + 2.82·43-s − 6.92·47-s − 1.00·49-s − 2.44·53-s + 4.89·55-s − 8·59-s − 3.46·61-s + 8.48·65-s + 11.3·67-s − 13.8·71-s − 4.89·77-s + 2.44·79-s − 14·83-s − 6.92·85-s + ⋯ |
| L(s) = 1 | − 1.09·5-s + 0.925·7-s − 0.603·11-s − 0.960·13-s + 0.685·17-s − 0.648·19-s + 1.44·23-s + 0.199·25-s − 0.454·29-s + 1.31·31-s − 1.01·35-s + 1.70·37-s − 1.32·41-s + 0.431·43-s − 1.01·47-s − 0.142·49-s − 0.336·53-s + 0.660·55-s − 1.04·59-s − 0.443·61-s + 1.05·65-s + 1.38·67-s − 1.64·71-s − 0.558·77-s + 0.275·79-s − 1.53·83-s − 0.751·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.977970846036624441728342257933, −7.43891196669451088017981050537, −6.67703548273872144142760343136, −5.61255603444409083591460822031, −4.76303966441167731508231180559, −4.44139873447408908647332969245, −3.29466308129427749025129874022, −2.54291559913886015923778399527, −1.29688258942090094575677826234, 0,
1.29688258942090094575677826234, 2.54291559913886015923778399527, 3.29466308129427749025129874022, 4.44139873447408908647332969245, 4.76303966441167731508231180559, 5.61255603444409083591460822031, 6.67703548273872144142760343136, 7.43891196669451088017981050537, 7.977970846036624441728342257933
