| L(s) = 1 | + 5.29·3-s + 5.29·5-s − 7·7-s + 19.0·9-s + 5.29·13-s + 28.0·15-s − 37.0·19-s − 37.0·21-s + 10·23-s + 3.00·25-s + 52.9·27-s − 37.0·35-s + 28.0·39-s + 100.·45-s + 49·49-s − 196.·57-s + 89.9·59-s − 121.·61-s − 133.·63-s + 28.0·65-s + 52.9·69-s − 110·71-s + 15.8·75-s + 130·79-s + 109.·81-s − 164.·83-s − 37.0·91-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 1.05·5-s − 7-s + 2.11·9-s + 0.407·13-s + 1.86·15-s − 1.94·19-s − 1.76·21-s + 0.434·23-s + 0.120·25-s + 1.95·27-s − 1.05·35-s + 0.717·39-s + 2.23·45-s + 0.999·49-s − 3.43·57-s + 1.52·59-s − 1.99·61-s − 2.11·63-s + 0.430·65-s + 0.766·69-s − 1.54·71-s + 0.211·75-s + 1.64·79-s + 1.34·81-s − 1.97·83-s − 0.407·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.865796818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865796818\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 5.29T + 9T^{2} \) |
| 5 | \( 1 - 5.29T + 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 5.29T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 37.0T + 361T^{2} \) |
| 23 | \( 1 - 10T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 89.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 121.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 110T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 130T + 6.24e3T^{2} \) |
| 83 | \( 1 + 164.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−12.54593275777135248413608703222, −10.63107339659927075728945767139, −9.820871395270317508559350665472, −9.075740617691953380863584004975, −8.364320728534320637291179143333, −7.02483034806797460800093035712, −6.04733758296183264534730937695, −4.18176101961362847268098920125, −2.97919793640807490832921116486, −1.94488605573529634134631247496,
1.94488605573529634134631247496, 2.97919793640807490832921116486, 4.18176101961362847268098920125, 6.04733758296183264534730937695, 7.02483034806797460800093035712, 8.364320728534320637291179143333, 9.075740617691953380863584004975, 9.820871395270317508559350665472, 10.63107339659927075728945767139, 12.54593275777135248413608703222
