| L(s) = 1 | − 1.84·5-s − 2i·11-s − 4.46i·13-s + 2.29·17-s − 1.53i·19-s − 8.82i·23-s − 1.58·25-s − 1.17i·29-s + 5.86i·31-s + 8.24·37-s + 11.8·41-s − 1.17·43-s + 8.02·47-s + 3.75i·53-s + 3.69i·55-s + ⋯ |
| L(s) = 1 | − 0.826·5-s − 0.603i·11-s − 1.23i·13-s + 0.556·17-s − 0.351i·19-s − 1.84i·23-s − 0.317·25-s − 0.217i·29-s + 1.05i·31-s + 1.35·37-s + 1.85·41-s − 0.178·43-s + 1.17·47-s + 0.516i·53-s + 0.498i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9229558969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9229558969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 4.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 + 1.53iT - 19T^{2} \) |
| 23 | \( 1 + 8.82iT - 23T^{2} \) |
| 29 | \( 1 + 1.17iT - 29T^{2} \) |
| 31 | \( 1 - 5.86iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 - 3.75iT - 53T^{2} \) |
| 59 | \( 1 + 9.81T + 59T^{2} \) |
| 61 | \( 1 - 12.3iT - 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 2.74iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 2.74iT - 97T^{2} \) |
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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
−7.64492113069555255428484829888, −7.18063557621026771891502651254, −6.02312406656873755014119887425, −5.77059291278314941491719250090, −4.58854719448539243226314584059, −4.16481350387712201762734185546, −3.05728375841168348549887866496, −2.69329154210325933276248202910, −1.11048373262862587614563108690, −0.26381859171003154244324912129,
1.22338871185203330208887761231, 2.14238445368566560764973976741, 3.18459500820480824902242495542, 4.14069778844501206337365624389, 4.32111172998016143829455674579, 5.50538668362902318993464385900, 6.07138360140644291079797763995, 7.02878691148839661643352002820, 7.65189290804866318124545974700, 7.905193241847409425341347401153
