| L(s) = 1 | + 2.71·3-s − 1.51·5-s + 2.20·7-s + 4.35·9-s − 6.02·11-s + 4.45·13-s − 4.11·15-s − 3.55·17-s − 3.54·19-s + 5.98·21-s + 4.23·23-s − 2.69·25-s + 3.66·27-s + 1.06·29-s − 6.50·31-s − 16.3·33-s − 3.35·35-s − 10.6·37-s + 12.0·39-s + 6.73·41-s − 10.4·43-s − 6.60·45-s − 5.48·47-s − 2.13·49-s − 9.63·51-s − 5.47·53-s + 9.15·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 0.679·5-s + 0.834·7-s + 1.45·9-s − 1.81·11-s + 1.23·13-s − 1.06·15-s − 0.862·17-s − 0.814·19-s + 1.30·21-s + 0.883·23-s − 0.538·25-s + 0.705·27-s + 0.198·29-s − 1.16·31-s − 2.84·33-s − 0.566·35-s − 1.75·37-s + 1.93·39-s + 1.05·41-s − 1.59·43-s − 0.985·45-s − 0.800·47-s − 0.304·49-s − 1.34·51-s − 0.751·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 - 2.20T + 7T^{2} \) |
| 11 | \( 1 + 6.02T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 6.50T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 - 9.11T + 59T^{2} \) |
| 61 | \( 1 + 5.04T + 61T^{2} \) |
| 67 | \( 1 - 0.260T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.93T + 97T^{2} \) |
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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
−7.66855961777493776072248061131, −7.18955975361674428706293752617, −6.18325014732047983093842764086, −5.13247350255591735826856431966, −4.57964676716948930882562086017, −3.66361169181885285221469791733, −3.19325165717433475159975183874, −2.23890783874033229198464146314, −1.65298861283566666418922701865, 0,
1.65298861283566666418922701865, 2.23890783874033229198464146314, 3.19325165717433475159975183874, 3.66361169181885285221469791733, 4.57964676716948930882562086017, 5.13247350255591735826856431966, 6.18325014732047983093842764086, 7.18955975361674428706293752617, 7.66855961777493776072248061131
