| L(s) = 1 | − 3.91·5-s − 4.32·7-s + 0.0827·11-s − 7.02·13-s + 5.10·17-s − 19-s + 7.83·23-s + 10.3·25-s + 2.81·29-s − 6·31-s + 16.9·35-s + 4·37-s − 1.02·41-s + 0.324·43-s + 6.73·47-s + 11.6·49-s − 8.85·53-s − 0.324·55-s − 14.0·59-s + 9.34·61-s + 27.5·65-s + 1.62·67-s − 1.62·71-s + 3.30·73-s − 0.357·77-s − 13.0·79-s + 6.81·83-s + ⋯ |
| L(s) = 1 | − 1.75·5-s − 1.63·7-s + 0.0249·11-s − 1.94·13-s + 1.23·17-s − 0.229·19-s + 1.63·23-s + 2.06·25-s + 0.522·29-s − 1.07·31-s + 2.86·35-s + 0.657·37-s − 0.159·41-s + 0.0494·43-s + 0.981·47-s + 1.67·49-s − 1.21·53-s − 0.0436·55-s − 1.82·59-s + 1.19·61-s + 3.41·65-s + 0.198·67-s − 0.193·71-s + 0.386·73-s − 0.0407·77-s − 1.46·79-s + 0.747·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5821550907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5821550907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 0.0827T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 - 0.324T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 9.34T + 61T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 + 5.66T + 97T^{2} \) |
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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
−9.572063408859655845484652905892, −8.864857151042532939187296599211, −7.66350081846257388068643578056, −7.34180215841084529654388967502, −6.55571372565931226527564508407, −5.28266169079199554620585240375, −4.38544768941895674888315408645, −3.35919795389486460116837587465, −2.84879744896194175781384474665, −0.52414676223566997405965517740,
0.52414676223566997405965517740, 2.84879744896194175781384474665, 3.35919795389486460116837587465, 4.38544768941895674888315408645, 5.28266169079199554620585240375, 6.55571372565931226527564508407, 7.34180215841084529654388967502, 7.66350081846257388068643578056, 8.864857151042532939187296599211, 9.572063408859655845484652905892
