| L(s) = 1 | − 20.7·3-s − 35.2·7-s + 186.·9-s − 7.33·11-s − 619.·13-s − 959.·17-s − 309.·19-s + 731.·21-s − 2.46e3·23-s + 1.16e3·27-s − 1.28e3·29-s + 7.09e3·31-s + 152.·33-s + 6.10e3·37-s + 1.28e4·39-s − 1.88e4·41-s − 3.14e3·43-s − 2.05e4·47-s − 1.55e4·49-s + 1.98e4·51-s − 3.37e4·53-s + 6.41e3·57-s + 1.50e4·59-s − 7.54e3·61-s − 6.59e3·63-s − 2.55e4·67-s + 5.11e4·69-s + ⋯ |
| L(s) = 1 | − 1.33·3-s − 0.272·7-s + 0.768·9-s − 0.0182·11-s − 1.01·13-s − 0.805·17-s − 0.196·19-s + 0.361·21-s − 0.972·23-s + 0.307·27-s − 0.283·29-s + 1.32·31-s + 0.0242·33-s + 0.732·37-s + 1.35·39-s − 1.74·41-s − 0.259·43-s − 1.35·47-s − 0.925·49-s + 1.07·51-s − 1.64·53-s + 0.261·57-s + 0.563·59-s − 0.259·61-s − 0.209·63-s − 0.696·67-s + 1.29·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3042519373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3042519373\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 20.7T + 243T^{2} \) |
| 7 | \( 1 + 35.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 7.33T + 1.61e5T^{2} \) |
| 13 | \( 1 + 619.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 959.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 309.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.10e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.54e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.76e5T + 8.58e9T^{2} \) |
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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
−9.874790033157485622027932310764, −8.671971242314746112117220770358, −7.66802099337573054339674527913, −6.60367845335515339914713528591, −6.15167978841141094017149483138, −5.03221852798659721024337832714, −4.45135699150610853096000669413, −2.99752783999240631088932623178, −1.70618328163818760779793480770, −0.26367646578635856089345607090,
0.26367646578635856089345607090, 1.70618328163818760779793480770, 2.99752783999240631088932623178, 4.45135699150610853096000669413, 5.03221852798659721024337832714, 6.15167978841141094017149483138, 6.60367845335515339914713528591, 7.66802099337573054339674527913, 8.671971242314746112117220770358, 9.874790033157485622027932310764
