| L(s) = 1 | + 8.18·2-s − 3.55·3-s + 34.9·4-s + 25·5-s − 29.1·6-s − 238.·7-s + 23.9·8-s − 230.·9-s + 204.·10-s + 121·11-s − 124.·12-s + 229.·13-s − 1.95e3·14-s − 88.9·15-s − 921.·16-s − 93.8·17-s − 1.88e3·18-s + 361·19-s + 873.·20-s + 849.·21-s + 989.·22-s − 1.10e3·23-s − 85.3·24-s + 625·25-s + 1.87e3·26-s + 1.68e3·27-s − 8.34e3·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 0.228·3-s + 1.09·4-s + 0.447·5-s − 0.330·6-s − 1.84·7-s + 0.132·8-s − 0.947·9-s + 0.646·10-s + 0.301·11-s − 0.249·12-s + 0.376·13-s − 2.66·14-s − 0.102·15-s − 0.899·16-s − 0.0787·17-s − 1.37·18-s + 0.229·19-s + 0.488·20-s + 0.420·21-s + 0.436·22-s − 0.433·23-s − 0.0302·24-s + 0.200·25-s + 0.544·26-s + 0.444·27-s − 2.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.767410977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767410977\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 8.18T + 32T^{2} \) |
| 3 | \( 1 + 3.55T + 243T^{2} \) |
| 7 | \( 1 + 238.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 229.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 93.8T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 849.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.15e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.66e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.01e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.95e4T + 8.58e9T^{2} \) |
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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
−9.343764758029616500490716772360, −8.440712437653149992808541985194, −6.94055862424762558928480315620, −6.25262978330978084225013622699, −5.90770733171622558217541717967, −4.94939683115632869890664177818, −3.77070767889313575761934446648, −3.16433550199670104851050463875, −2.34622232204145269692126973428, −0.54776394743374383083129262449,
0.54776394743374383083129262449, 2.34622232204145269692126973428, 3.16433550199670104851050463875, 3.77070767889313575761934446648, 4.94939683115632869890664177818, 5.90770733171622558217541717967, 6.25262978330978084225013622699, 6.94055862424762558928480315620, 8.440712437653149992808541985194, 9.343764758029616500490716772360
