| L(s) = 1 | − 2.21i·3-s + (−1.81 + 1.30i)5-s + (−1.09 − 2.41i)7-s − 1.90·9-s + 0.661i·11-s + 0.235·13-s + (2.88 + 4.02i)15-s + 6.02·17-s + 7.98·19-s + (−5.33 + 2.41i)21-s + 8.48·23-s + (1.61 − 4.73i)25-s − 2.43i·27-s − 7.86·29-s + 1.36·31-s + ⋯ |
| L(s) = 1 | − 1.27i·3-s + (−0.813 + 0.581i)5-s + (−0.412 − 0.910i)7-s − 0.633·9-s + 0.199i·11-s + 0.0652·13-s + (0.743 + 1.03i)15-s + 1.46·17-s + 1.83·19-s + (−1.16 + 0.527i)21-s + 1.76·23-s + (0.322 − 0.946i)25-s − 0.468i·27-s − 1.46·29-s + 0.245·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.439785256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439785256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.81 - 1.30i)T \) |
| 7 | \( 1 + (1.09 + 2.41i)T \) |
| good | 3 | \( 1 + 2.21iT - 3T^{2} \) |
| 11 | \( 1 - 0.661iT - 11T^{2} \) |
| 13 | \( 1 - 0.235T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 + 2.79iT - 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 7.72iT - 47T^{2} \) |
| 53 | \( 1 + 8.45iT - 53T^{2} \) |
| 59 | \( 1 + 0.929T + 59T^{2} \) |
| 61 | \( 1 + 2.28iT - 61T^{2} \) |
| 67 | \( 1 + 3.19T + 67T^{2} \) |
| 71 | \( 1 + 0.619iT - 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 7.01iT - 79T^{2} \) |
| 83 | \( 1 + 7.05iT - 83T^{2} \) |
| 89 | \( 1 + 9.97iT - 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−8.465135277759339768096659440273, −7.67798036989795336521178434834, −7.16547554990545569803823443576, −6.92976002564943382763386239343, −5.79602929536439580408406228903, −4.80690210691987545588209366456, −3.45284735183265271496665183259, −3.13719146986916297610451012901, −1.53252115157076829970058254985, −0.61260879399516060958884251151,
1.17093065018352409477041047492, 3.19064549348067972257345580609, 3.34708333668199791292482963738, 4.53572908372658709919652614756, 5.30991283937594167460982992607, 5.70015937433412096833689708711, 7.17906663450146678476235169476, 7.80305136583985340107761738290, 8.821819107350274322609761708232, 9.330292119615800120759975250216
