| L(s) = 1 | + (−1.65 − 2.5i)3-s + (−3.5 + 8.29i)9-s − 16.5i·11-s + 10i·13-s − 3.31·17-s + 7·19-s − 19.8·23-s + (26.5 − 4.99i)27-s − 33.1i·29-s − 42·31-s + (−41.4 + 27.5i)33-s + 40i·37-s + (25 − 16.5i)39-s − 16.5i·41-s + 50i·43-s + ⋯ |
| L(s) = 1 | + (−0.552 − 0.833i)3-s + (−0.388 + 0.921i)9-s − 1.50i·11-s + 0.769i·13-s − 0.195·17-s + 0.368·19-s − 0.865·23-s + (0.982 − 0.185i)27-s − 1.14i·29-s − 1.35·31-s + (−1.25 + 0.833i)33-s + 1.08i·37-s + (0.641 − 0.425i)39-s − 0.404i·41-s + 1.16i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3382985324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3382985324\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 + 2.5i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 16.5iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 + 3.31T + 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 + 19.8T + 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 + 42T + 961T^{2} \) |
| 37 | \( 1 - 40iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 46.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 46.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 45iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 35iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 12T + 6.24e3T^{2} \) |
| 83 | \( 1 + 69.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 70iT - 9.40e3T^{2} \) |
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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
−9.790314688207897227118917340284, −8.793632533384320382566809262215, −8.083540013664912652169579814680, −7.29437840939081650223873499196, −6.27659817678836439021130886047, −5.86343174535615408539281897524, −4.77471490639852831750728324547, −3.57234188002098584414905984081, −2.35648835150368943190261333084, −1.16066254331973412799473948018,
0.11540378507972748670793894992, 1.84882816263339239707993151531, 3.27032738525696349516933458369, 4.19833468788243842648382983305, 5.07402018450261484579416632750, 5.74809075835268551589423318747, 6.85726128012539186852687018220, 7.58483952876394651399984037266, 8.724638445955613863402311429212, 9.525974134692243406140624646587
