| L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s − 2.35·7-s − i·8-s + 9-s + 10-s − 2.04·11-s − 12-s + 3.54i·13-s − 2.35i·14-s − i·15-s + 16-s + 0.356i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.890·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.616·11-s − 0.288·12-s + 0.982i·13-s − 0.629i·14-s − 0.258i·15-s + 0.250·16-s + 0.0863i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093296523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093296523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (4.94 + 3.54i)T \) |
| good | 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 3.54iT - 13T^{2} \) |
| 17 | \( 1 - 0.356iT - 17T^{2} \) |
| 19 | \( 1 - 5.94iT - 19T^{2} \) |
| 23 | \( 1 - 3.54iT - 23T^{2} \) |
| 29 | \( 1 - 7.99iT - 29T^{2} \) |
| 31 | \( 1 - 3.18iT - 31T^{2} \) |
| 41 | \( 1 + 3.18T + 41T^{2} \) |
| 43 | \( 1 - 3.89iT - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.356T + 53T^{2} \) |
| 59 | \( 1 - 7.08iT - 59T^{2} \) |
| 61 | \( 1 + 6.30iT - 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 + 0.712T + 71T^{2} \) |
| 73 | \( 1 + 6.25T + 73T^{2} \) |
| 79 | \( 1 - 0.372iT - 79T^{2} \) |
| 83 | \( 1 - 7.63T + 83T^{2} \) |
| 89 | \( 1 + 9.54iT - 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−9.927954084812183927044465988095, −9.149528010708865446260622534965, −8.607094238208894819218933764421, −7.63989599728036987558224596361, −6.95836020322287513750077434220, −6.00690200997084440844341542632, −5.13193821683996462057190819753, −4.02547163112643252288104562762, −3.18997448682146557318369095360, −1.64349430186259516883569783444,
0.43816467838509020214220256905, 2.40818767875137690903752965027, 2.93937626845352026859992847508, 3.92330080889104943902022518475, 5.04839364317978323270586977772, 6.14205064666038480903044424994, 7.11830811438907212827010152247, 7.992854112431506850183403838290, 8.832471828404299077009877713759, 9.675400608699430188452591518282
