| L(s) = 1 | − 3.46·5-s + 7-s − 1.46·11-s + 2·13-s − 0.535·17-s − 6.92·19-s − 1.46·23-s + 6.99·25-s + 4.92·29-s + 10.9·31-s − 3.46·35-s − 2·37-s − 11.4·41-s + 8·43-s + 10.9·47-s + 49-s + 2·53-s + 5.07·55-s − 1.07·59-s − 8.92·61-s − 6.92·65-s + 2.92·67-s + 9.46·71-s + 12.9·73-s − 1.46·77-s + 10.9·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 1.54·5-s + 0.377·7-s − 0.441·11-s + 0.554·13-s − 0.129·17-s − 1.58·19-s − 0.305·23-s + 1.39·25-s + 0.915·29-s + 1.96·31-s − 0.585·35-s − 0.328·37-s − 1.79·41-s + 1.21·43-s + 1.59·47-s + 0.142·49-s + 0.274·53-s + 0.683·55-s − 0.139·59-s − 1.14·61-s − 0.859·65-s + 0.357·67-s + 1.12·71-s + 1.51·73-s − 0.166·77-s + 1.22·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118381211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118381211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.753377106840359325550546439270, −8.358734294957248580673881981374, −7.77590251642253916492479492068, −6.86360148088145715715249238443, −6.11065955209596679955876341726, −4.82662388354012588224526323501, −4.28065559316850229570490989414, −3.43559858055096352650808262398, −2.30489021656926531144153411702, −0.69259820703426237827432235039,
0.69259820703426237827432235039, 2.30489021656926531144153411702, 3.43559858055096352650808262398, 4.28065559316850229570490989414, 4.82662388354012588224526323501, 6.11065955209596679955876341726, 6.86360148088145715715249238443, 7.77590251642253916492479492068, 8.358734294957248580673881981374, 8.753377106840359325550546439270
