| L(s) = 1 | + 1.59·5-s + 1.16i·7-s − 1.98i·11-s − 0.164i·13-s + 3.57i·17-s + 3.16·19-s + 1.21·23-s − 2.44·25-s + 4.98·29-s − 6.64i·31-s + 1.85i·35-s + 1.10i·37-s + 2.02i·41-s + 8.34·43-s − 2.21·47-s + ⋯ |
| L(s) = 1 | + 0.714·5-s + 0.440i·7-s − 0.597i·11-s − 0.0455i·13-s + 0.867i·17-s + 0.725·19-s + 0.253·23-s − 0.489·25-s + 0.925·29-s − 1.19i·31-s + 0.314i·35-s + 0.181i·37-s + 0.316i·41-s + 1.27·43-s − 0.323·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.331852755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331852755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 1.16iT - 7T^{2} \) |
| 11 | \( 1 + 1.98iT - 11T^{2} \) |
| 13 | \( 1 + 0.164iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 + 6.64iT - 31T^{2} \) |
| 37 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 2.02iT - 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 0.888iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.879T + 73T^{2} \) |
| 79 | \( 1 + 2.91iT - 79T^{2} \) |
| 83 | \( 1 + 13.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.41iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.193581404023602067077185184941, −7.67449513966416650517454108347, −6.62867004609732029989131736253, −5.97847190619511519920930666695, −5.55646674048077171474104682313, −4.61259566495934234782872495592, −3.69684613502032844671309350027, −2.79672347019827950222043487765, −1.99156023458084062213281151016, −0.897695623420779653313694258016,
0.800987921106885069141632984949, 1.85291591291755721933988230716, 2.75488713780447143108221926333, 3.63686432990809203143293497138, 4.65798079069937062988985451195, 5.18902417944002685955752438909, 6.03607346403273660204416087668, 6.83881337884632543934075966770, 7.34710856949679658925211332329, 8.130575485642006195231907886743
