| L(s) = 1 | + 7.31i·2-s + 202.·4-s + 3.66e3·7-s + 3.35e3i·8-s − 1.32e4i·11-s − 2.16e4·13-s + 2.67e4i·14-s + 2.73e4·16-s + 1.17e5i·17-s − 2.55e5·19-s + 9.69e4·22-s + 4.21e5i·23-s − 1.58e5i·26-s + 7.41e5·28-s + 6.10e5i·29-s + ⋯ |
| L(s) = 1 | + 0.457i·2-s + 0.791·4-s + 1.52·7-s + 0.818i·8-s − 0.905i·11-s − 0.756·13-s + 0.697i·14-s + 0.416·16-s + 1.40i·17-s − 1.96·19-s + 0.414·22-s + 1.50i·23-s − 0.345i·26-s + 1.20·28-s + 0.863i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.467733722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.467733722\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7.31iT - 256T^{2} \) |
| 7 | \( 1 - 3.66e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.32e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.16e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.17e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.55e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.21e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.10e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.80e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.00e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.06e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.60e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.57e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.30e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.09e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.16e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.46e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.44e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.16e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.05e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.84e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.57e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 6.01e7T + 7.83e15T^{2} \) |
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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
−10.99203419285273561210788318302, −10.58657120339881528649053488556, −8.788896847534650334210504535284, −8.100533676462728116716728337735, −7.24149096992548090724248584841, −6.03372389122263212534419692485, −5.20415086465459817172106175312, −3.85318656551523089780068034711, −2.24524277295708553630349797813, −1.44315376023335746282438317402,
0.47423763267497606008745867192, 1.96721858960230044982169595463, 2.40025529579395179064015570919, 4.22133656774521707097911621014, 5.04957807464713061007033607428, 6.58987572033885359898719803814, 7.43888582887554580793784616922, 8.398206050112833577395134884751, 9.720223852507551186095045781489, 10.67200774961078738025763679397
