| L(s) = 1 | − 2.64i·2-s − 3.00·4-s + 11.2·7-s − 2.64i·8-s − 4.24i·11-s + 11.2·13-s − 29.6i·14-s − 18.9·16-s − 10.5i·17-s − 20·19-s − 11.2·22-s − 5.29i·23-s − 29.6i·26-s − 33.6·28-s − 8.48i·29-s + ⋯ |
| L(s) = 1 | − 1.32i·2-s − 0.750·4-s + 1.60·7-s − 0.330i·8-s − 0.385i·11-s + 0.863·13-s − 2.12i·14-s − 1.18·16-s − 0.622i·17-s − 1.05·19-s − 0.510·22-s − 0.230i·23-s − 1.14i·26-s − 1.20·28-s − 0.292i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.837380 - 1.61769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.837380 - 1.61769i\) |
| \(L(2)\) |
|
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 | \( 1 + 2.64iT - 4T^{2} \) |
| 7 | \( 1 - 11.2T + 49T^{2} \) |
| 11 | \( 1 + 4.24iT - 121T^{2} \) |
| 13 | \( 1 - 11.2T + 169T^{2} \) |
| 17 | \( 1 + 10.5iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 5.29iT - 529T^{2} \) |
| 29 | \( 1 + 8.48iT - 841T^{2} \) |
| 31 | \( 1 - 26T + 961T^{2} \) |
| 37 | \( 1 + 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 55.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 22.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 46.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 67.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 14T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.4T + 9.40e3T^{2} \) |
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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
−11.38020456201942924699480149328, −11.07251277493000026820162927007, −10.08898216862017590169412023573, −8.823591286066384544964609765432, −8.025600263840239737755999330761, −6.52617123230284920621000946189, −4.96735298161233952164142898068, −3.91285342762432973155287660059, −2.40392633734657348304349419978, −1.14804305772938842096207146812,
1.88075454730649538828532941852, 4.23739862473275538878131488710, 5.24511381819518476455370923540, 6.27548230695885076146624175486, 7.38161315369540023169646876468, 8.272656334490200088131503145574, 8.829886369583236264775668643586, 10.56704401917107635811286803538, 11.27326738733517251479102014527, 12.40125826073179931096854080867
