| L(s) = 1 | − 2.73i·3-s − i·5-s + 4.73·7-s − 4.46·9-s − 3.46i·11-s + 3.46i·13-s − 2.73·15-s − 3.46·17-s + 2i·19-s − 12.9i·21-s − 2.19·23-s − 25-s + 3.99i·27-s − 2.53·31-s − 9.46·33-s + ⋯ |
| L(s) = 1 | − 1.57i·3-s − 0.447i·5-s + 1.78·7-s − 1.48·9-s − 1.04i·11-s + 0.960i·13-s − 0.705·15-s − 0.840·17-s + 0.458i·19-s − 2.82i·21-s − 0.457·23-s − 0.200·25-s + 0.769i·27-s − 0.455·31-s − 1.64·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.891812 - 1.16223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891812 - 1.16223i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.73iT - 3T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196iT - 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 - 0.196iT - 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−11.51619100062475453597768216747, −10.91456290682788660469075623065, −9.070321189516178084666430431450, −8.262800374752680871730734032005, −7.71261604603941745553429114516, −6.57404807877663760465410882540, −5.55839792909619361598525833106, −4.30954755324240951155302494286, −2.20992760395063336859024110249, −1.22447303076733250374926377475,
2.31690934466756140279194415739, 3.97851902355099256593992072877, 4.74923742536354138952265870981, 5.56678722007914872470340498459, 7.28482561540555240027552186332, 8.260522407781695279026341291730, 9.238621253436707852103916216737, 10.19976779368031626323688120447, 10.93074474883918805510494030016, 11.41252855123899983986942115017
