| L(s) = 1 | + 1.57·3-s + 25·5-s + 49·7-s − 240.·9-s + 322.·11-s + 657.·13-s + 39.4·15-s + 665.·17-s − 2.21e3·19-s + 77.3·21-s + 1.58e3·23-s + 625·25-s − 763.·27-s + 6.38e3·29-s + 7.71e3·31-s + 509.·33-s + 1.22e3·35-s + 3.43e3·37-s + 1.03e3·39-s − 6.53e3·41-s + 1.66e4·43-s − 6.01e3·45-s + 1.25e4·47-s + 2.40e3·49-s + 1.05e3·51-s + 4.24e3·53-s + 8.06e3·55-s + ⋯ |
| L(s) = 1 | + 0.101·3-s + 0.447·5-s + 0.377·7-s − 0.989·9-s + 0.804·11-s + 1.07·13-s + 0.0452·15-s + 0.558·17-s − 1.40·19-s + 0.0382·21-s + 0.625·23-s + 0.200·25-s − 0.201·27-s + 1.41·29-s + 1.44·31-s + 0.0814·33-s + 0.169·35-s + 0.412·37-s + 0.109·39-s − 0.607·41-s + 1.37·43-s − 0.442·45-s + 0.826·47-s + 0.142·49-s + 0.0565·51-s + 0.207·53-s + 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.298602343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298602343\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 1.57T + 243T^{2} \) |
| 11 | \( 1 - 322.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 657.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 665.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.21e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.15e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5T + 8.58e9T^{2} \) |
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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
−12.15039991615023284069594071000, −11.22629832198637778811628741629, −10.26603845214062733686387968432, −8.885996551316146855696088528458, −8.311407323172569179001205108894, −6.63160997767520776506860706389, −5.73927544067276422839766454541, −4.23520268526526143041820175965, −2.70761496771098044593245423352, −1.08545209538125062075450240043,
1.08545209538125062075450240043, 2.70761496771098044593245423352, 4.23520268526526143041820175965, 5.73927544067276422839766454541, 6.63160997767520776506860706389, 8.311407323172569179001205108894, 8.885996551316146855696088528458, 10.26603845214062733686387968432, 11.22629832198637778811628741629, 12.15039991615023284069594071000
