| L(s) = 1 | − 11.4i·3-s + 39.4i·7-s + 111.·9-s − 121·11-s + 1.18e3i·13-s − 2.19e3i·17-s − 1.86e3·19-s + 452.·21-s − 337. i·23-s − 4.06e3i·27-s + 5.43e3·29-s + 139.·31-s + 1.38e3i·33-s + 5.94e3i·37-s + 1.35e4·39-s + ⋯ |
| L(s) = 1 | − 0.736i·3-s + 0.303i·7-s + 0.457·9-s − 0.301·11-s + 1.94i·13-s − 1.84i·17-s − 1.18·19-s + 0.223·21-s − 0.133i·23-s − 1.07i·27-s + 1.19·29-s + 0.0260·31-s + 0.222i·33-s + 0.713i·37-s + 1.42·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.495666540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495666540\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 11.4iT - 243T^{2} \) |
| 7 | \( 1 - 39.4iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 1.18e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.19e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 337. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 139.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.94e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.41e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.65e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.96e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.16e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.62e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.88e4iT - 8.58e9T^{2} \) |
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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
−9.215525648740368380606469758025, −8.475260948014453464421385932999, −7.42723736850743746645716444086, −6.81901904183637951233257218490, −6.19842363994151196010538140998, −4.82462188272964759421427834055, −4.26193077337945109707428935856, −2.73146857605401325412102011246, −1.98420852725155193477700238157, −0.932995029640330184735261941202,
0.31192322055350848010806043173, 1.52626091695872239114320824093, 2.85097396759804637067713243550, 3.82411314157394193909290391569, 4.52089667855864778015685285805, 5.56250393273855770003121064106, 6.30358287705083717695681325308, 7.49371227597519187195961149195, 8.196201185228477294043659081357, 8.946380396210580849151096313167
