| L(s) = 1 | − 2·4-s + 4.12i·5-s + 4.12i·11-s − 13-s + 4·16-s − 4.12i·17-s + 5·19-s − 8.24i·20-s + 4.12i·23-s − 12·25-s − 8.24i·29-s + 4.12i·41-s + 11·43-s − 8.24i·44-s + 7·49-s + ⋯ |
| L(s) = 1 | − 4-s + 1.84i·5-s + 1.24i·11-s − 0.277·13-s + 16-s − 0.999i·17-s + 1.14·19-s − 1.84i·20-s + 0.859i·23-s − 2.40·25-s − 1.53i·29-s + 0.643i·41-s + 1.67·43-s − 1.24i·44-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.602549 + 0.602549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602549 + 0.602549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 4.12iT \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 4.12iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4.12iT - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4.12iT - 23T^{2} \) |
| 29 | \( 1 + 8.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.12iT - 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 16.4iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−13.53775892560318106310910398985, −12.15041026925029490552105856456, −11.20302199190088921731449458790, −9.894078556665582167550913173828, −9.603643811261349093777838647689, −7.72407406514636345892186587754, −7.07083890770692774176396054804, −5.58995675492137965916450338420, −4.11086895909721362941669626921, −2.72129187565145507994058986416,
0.931369063223970506871556425315, 3.79280369284563752828600486905, 4.97419961464472443978675598258, 5.77231813928881131795249053068, 7.906924833504903118323478670203, 8.756006203078045934159160170532, 9.262694930711212273657472698289, 10.60469007478186818262450510920, 12.12811474436563540864673653366, 12.73771041920417490052266237753
