| L(s) = 1 | − 1.56·2-s − 8.24·3-s − 5.56·4-s − 5·5-s + 12.8·6-s + 33.1·7-s + 21.1·8-s + 41·9-s + 7.80·10-s − 47.1·11-s + 45.8·12-s − 13·13-s − 51.7·14-s + 41.2·15-s + 11.4·16-s + 51.9·17-s − 64.0·18-s + 37.6·19-s + 27.8·20-s − 273.·21-s + 73.5·22-s + 161.·23-s − 174.·24-s + 25·25-s + 20.3·26-s − 115.·27-s − 184.·28-s + ⋯ |
| L(s) = 1 | − 0.552·2-s − 1.58·3-s − 0.695·4-s − 0.447·5-s + 0.876·6-s + 1.78·7-s + 0.935·8-s + 1.51·9-s + 0.246·10-s − 1.29·11-s + 1.10·12-s − 0.277·13-s − 0.987·14-s + 0.709·15-s + 0.178·16-s + 0.741·17-s − 0.838·18-s + 0.454·19-s + 0.310·20-s − 2.83·21-s + 0.713·22-s + 1.46·23-s − 1.48·24-s + 0.200·25-s + 0.153·26-s − 0.822·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5465293884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5465293884\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 3 | \( 1 + 8.24T + 27T^{2} \) |
| 7 | \( 1 - 33.1T + 343T^{2} \) |
| 11 | \( 1 + 47.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.27T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 53.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 395.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 367.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 442T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.09e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 722.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 114.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 806.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.73e3T + 9.12e5T^{2} \) |
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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
−14.46048828321443275086824389706, −13.05899622551236666380601128273, −11.82566750241983360190911898319, −10.99760295427020900573047153306, −10.14665464535187641885795729219, −8.330754306876171201567212720764, −7.42886498105223520283804301391, −5.25608904272152771257816933963, −4.77897763965871969265766141467, −0.902210881104670927154845889665,
0.902210881104670927154845889665, 4.77897763965871969265766141467, 5.25608904272152771257816933963, 7.42886498105223520283804301391, 8.330754306876171201567212720764, 10.14665464535187641885795729219, 10.99760295427020900573047153306, 11.82566750241983360190911898319, 13.05899622551236666380601128273, 14.46048828321443275086824389706
