| L(s) = 1 | + 3.37·2-s + 3·3-s + 3.37·4-s − 3.48·5-s + 10.1·6-s − 4.74·7-s − 15.6·8-s + 9·9-s − 11.7·10-s + 11·11-s + 10.1·12-s − 15.0·13-s − 16·14-s − 10.4·15-s − 79.6·16-s + 73.1·17-s + 30.3·18-s − 78.7·19-s − 11.7·20-s − 14.2·21-s + 37.0·22-s + 112·23-s − 46.8·24-s − 112.·25-s − 50.6·26-s + 27·27-s − 16·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.577·3-s + 0.421·4-s − 0.312·5-s + 0.688·6-s − 0.256·7-s − 0.689·8-s + 0.333·9-s − 0.372·10-s + 0.301·11-s + 0.243·12-s − 0.320·13-s − 0.305·14-s − 0.180·15-s − 1.24·16-s + 1.04·17-s + 0.397·18-s − 0.950·19-s − 0.131·20-s − 0.147·21-s + 0.359·22-s + 1.01·23-s − 0.398·24-s − 0.902·25-s − 0.382·26-s + 0.192·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.138345534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138345534\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 5 | \( 1 + 3.48T + 125T^{2} \) |
| 7 | \( 1 + 4.74T + 343T^{2} \) |
| 13 | \( 1 + 15.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 - 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 102.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 280.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196T + 2.05e5T^{2} \) |
| 61 | \( 1 + 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 900.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 756.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 756.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 614.T + 9.12e5T^{2} \) |
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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
−15.72197703177180852690966096402, −14.80222814541924372359390981129, −13.87430331569316197081791324882, −12.78081106325742402964963034825, −11.76900512853269557218188131478, −9.866108343323399937237525599678, −8.327450473960769027452790593869, −6.49310939166068878234154025260, −4.66921356206518403831190459786, −3.15974324457558183552847942675,
3.15974324457558183552847942675, 4.66921356206518403831190459786, 6.49310939166068878234154025260, 8.327450473960769027452790593869, 9.866108343323399937237525599678, 11.76900512853269557218188131478, 12.78081106325742402964963034825, 13.87430331569316197081791324882, 14.80222814541924372359390981129, 15.72197703177180852690966096402
