| L(s) = 1 | − 3.07·3-s + 3.88·5-s + 0.0828·7-s + 6.47·9-s − 0.885·11-s − 3.52·13-s − 11.9·15-s − 5.02·17-s + 5.34·19-s − 0.255·21-s − 5.11·23-s + 10.1·25-s − 10.6·27-s + 0.465·29-s + 1.81·31-s + 2.72·33-s + 0.322·35-s + 4.29·37-s + 10.8·39-s + 11.2·41-s + 1.13·43-s + 25.1·45-s − 3.32·47-s − 6.99·49-s + 15.4·51-s − 11.4·53-s − 3.44·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s + 1.73·5-s + 0.0313·7-s + 2.15·9-s − 0.266·11-s − 0.976·13-s − 3.09·15-s − 1.21·17-s + 1.22·19-s − 0.0556·21-s − 1.06·23-s + 2.02·25-s − 2.05·27-s + 0.0864·29-s + 0.325·31-s + 0.474·33-s + 0.0544·35-s + 0.706·37-s + 1.73·39-s + 1.76·41-s + 0.172·43-s + 3.75·45-s − 0.484·47-s − 0.999·49-s + 2.16·51-s − 1.57·53-s − 0.464·55-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 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 - 0.0828T + 7T^{2} \) |
| 11 | \( 1 + 0.885T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 - 0.465T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 4.14T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.21961711403053513560511911056, −6.47217358584252916397755552841, −6.05308933041108066910476015205, −5.54250203936419970524696496352, −4.83239376244092249813398302475, −4.42411481174817709565100477178, −2.86188443369719644854233255639, −2.01562992228348903182025909037, −1.19961996505316264928602563795, 0,
1.19961996505316264928602563795, 2.01562992228348903182025909037, 2.86188443369719644854233255639, 4.42411481174817709565100477178, 4.83239376244092249813398302475, 5.54250203936419970524696496352, 6.05308933041108066910476015205, 6.47217358584252916397755552841, 7.21961711403053513560511911056
