| L(s) = 1 | + 9·3-s − 9.73i·5-s − 105. i·7-s + 81·9-s − 269. i·11-s + (227. − 565. i)13-s − 87.6i·15-s + 1.66e3·17-s + 2.82i·19-s − 946. i·21-s − 2.15e3·23-s + 3.03e3·25-s + 729·27-s + 220.·29-s + 788. i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.174i·5-s − 0.811i·7-s + 0.333·9-s − 0.671i·11-s + (0.373 − 0.927i)13-s − 0.100i·15-s + 1.39·17-s + 0.00179i·19-s − 0.468i·21-s − 0.847·23-s + 0.969·25-s + 0.192·27-s + 0.0487·29-s + 0.147i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.500813392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500813392\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + (-227. + 565. i)T \) |
| good | 5 | \( 1 + 9.73iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 105. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 269. iT - 1.61e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.82iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 220.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 788. iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 980. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.48e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.51e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.50e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.44e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.35e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.58e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.81e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.62e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.26e5iT - 8.58e9T^{2} \) |
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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
−9.702959958826889378390510826417, −8.429783374204288222139686105920, −8.027760978500492930722920302208, −7.02588150096757444305266171911, −5.93645725721309801042552248668, −4.91592431606194350669327606818, −3.66743076767519357441118507892, −3.05000605660656356164343634279, −1.45750210164007715785495144194, −0.50701195193075504634761714042,
1.35578923093332468621247579187, 2.35851593384745927861754297867, 3.40371240542544688178072234317, 4.48709561893140109337709841220, 5.59506836407597667101801719221, 6.59672842417611076446066574968, 7.54593360723733782187989035426, 8.413533170859670448526809533543, 9.249422296702242575168032316724, 9.913490408961604823672302631685
