| L(s) = 1 | + (2.23 − 2i)3-s + 6i·7-s + (1.00 − 8.94i)9-s + 4.47i·11-s − 16i·13-s + 4.47·17-s − 2·19-s + (12 + 13.4i)21-s + 13.4·23-s + (−15.6 − 22.0i)27-s − 31.3i·29-s + 18·31-s + (8.94 + 10.0i)33-s − 16i·37-s + (−32 − 35.7i)39-s + ⋯ |
| L(s) = 1 | + (0.745 − 0.666i)3-s + 0.857i·7-s + (0.111 − 0.993i)9-s + 0.406i·11-s − 1.23i·13-s + 0.263·17-s − 0.105·19-s + (0.571 + 0.638i)21-s + 0.583·23-s + (−0.579 − 0.814i)27-s − 1.07i·29-s + 0.580·31-s + (0.271 + 0.303i)33-s − 0.432i·37-s + (−0.820 − 0.917i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.447315388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447315388\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.23 + 2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6iT - 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 16iT - 169T^{2} \) |
| 17 | \( 1 - 4.47T + 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 - 13.4T + 529T^{2} \) |
| 29 | \( 1 + 31.3iT - 841T^{2} \) |
| 31 | \( 1 - 18T + 961T^{2} \) |
| 37 | \( 1 + 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 4.47T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 138T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 166iT - 9.40e3T^{2} \) |
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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
−9.261264890818581470616682327406, −8.492867419307736634748209572353, −7.84007304666168912960373123496, −7.05353083454605931316078301303, −6.04804253610124767334540541956, −5.29537194593361926368684067240, −3.95516148501361182646853280115, −2.88280021473956159449404422373, −2.14576131624126011592852865662, −0.71287348438050975248604658101,
1.26863766900372015204362167603, 2.62847361549277701450153541401, 3.66543254198550618174653912190, 4.37775575061989648671811380127, 5.25982800382105491867861793039, 6.57949228244667172174811423115, 7.30840976485044757962857454494, 8.236607375941879159359517404678, 8.952546035451663062349726452289, 9.677054889230383227442479060262
