| L(s) = 1 | − 5-s + 1.45·7-s + 4.79·11-s + 13-s − 0.545·17-s + 7.33·19-s − 0.545·23-s + 25-s + 5.33·29-s + 2·31-s − 1.45·35-s − 3.88·37-s − 5.70·41-s + 2.90·43-s − 1.09·47-s − 4.88·49-s − 4.79·53-s − 4.79·55-s − 10.6·59-s + 7.88·61-s − 65-s + 3.09·67-s + 2.97·71-s − 4.42·73-s + 6.97·77-s + 11.7·79-s + 0.545·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.549·7-s + 1.44·11-s + 0.277·13-s − 0.132·17-s + 1.68·19-s − 0.113·23-s + 0.200·25-s + 0.991·29-s + 0.359·31-s − 0.245·35-s − 0.638·37-s − 0.890·41-s + 0.443·43-s − 0.159·47-s − 0.697·49-s − 0.658·53-s − 0.646·55-s − 1.38·59-s + 1.00·61-s − 0.124·65-s + 0.377·67-s + 0.352·71-s − 0.518·73-s + 0.794·77-s + 1.31·79-s + 0.0591·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.291499694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291499694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 17 | \( 1 + 0.545T + 17T^{2} \) |
| 19 | \( 1 - 7.33T + 19T^{2} \) |
| 23 | \( 1 + 0.545T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 - 3.09T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−8.257512416260129861949369968322, −7.64956568189119024722233676953, −6.82467850390337773067470170172, −6.27460505392401849062337172302, −5.24124398685439353920346166692, −4.60085340496128536252634903743, −3.72694505235637466041107259032, −3.07982816989251930468390253942, −1.72917243098590532607996005517, −0.909001220025816479099790727499,
0.909001220025816479099790727499, 1.72917243098590532607996005517, 3.07982816989251930468390253942, 3.72694505235637466041107259032, 4.60085340496128536252634903743, 5.24124398685439353920346166692, 6.27460505392401849062337172302, 6.82467850390337773067470170172, 7.64956568189119024722233676953, 8.257512416260129861949369968322
