| L(s) = 1 | + 2.93·3-s − 0.396·5-s + 4.33·7-s + 5.59·9-s + 3.96·11-s − 2.03·13-s − 1.16·15-s + 4.54·17-s + 0.578·19-s + 12.7·21-s − 6.24·23-s − 4.84·25-s + 7.61·27-s + 5.13·29-s − 0.0539·31-s + 11.6·33-s − 1.71·35-s − 0.864·37-s − 5.97·39-s − 0.878·41-s + 2.23·43-s − 2.21·45-s − 9.44·47-s + 11.7·49-s + 13.3·51-s + 10.8·53-s − 1.57·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s − 0.177·5-s + 1.63·7-s + 1.86·9-s + 1.19·11-s − 0.565·13-s − 0.299·15-s + 1.10·17-s + 0.132·19-s + 2.77·21-s − 1.30·23-s − 0.968·25-s + 1.46·27-s + 0.954·29-s − 0.00969·31-s + 2.02·33-s − 0.290·35-s − 0.142·37-s − 0.957·39-s − 0.137·41-s + 0.341·43-s − 0.330·45-s − 1.37·47-s + 1.68·49-s + 1.86·51-s + 1.49·53-s − 0.211·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.686338177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686338177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 + 0.396T + 5T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 0.578T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 5.13T + 29T^{2} \) |
| 31 | \( 1 + 0.0539T + 31T^{2} \) |
| 37 | \( 1 + 0.864T + 37T^{2} \) |
| 41 | \( 1 + 0.878T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 8.15T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 + 5.74T + 79T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 + 3.36T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.407271515608855854227618194943, −7.72586752650121921617223752582, −7.49630144378712936349132014366, −6.35431570687577995328886673450, −5.27442718668359835489751506650, −4.33681628257775466891130091630, −3.88393515280443493893373758482, −2.91583828991089330653270259432, −1.92661119008048369817781033253, −1.35546740724348926764193699752,
1.35546740724348926764193699752, 1.92661119008048369817781033253, 2.91583828991089330653270259432, 3.88393515280443493893373758482, 4.33681628257775466891130091630, 5.27442718668359835489751506650, 6.35431570687577995328886673450, 7.49630144378712936349132014366, 7.72586752650121921617223752582, 8.407271515608855854227618194943
