| L(s) = 1 | + 1.86·2-s + 2.47·4-s − 1.20·5-s + 2.75·8-s + 9-s − 2.24·10-s − 1.96·13-s + 2.66·16-s + 1.86·18-s + 0.184·19-s − 2.98·20-s + 0.452·25-s − 3.66·26-s − 1.70·29-s + 0.891·31-s + 2.20·32-s + 2.47·36-s + 0.344·38-s − 3.32·40-s − 1.70·43-s − 1.20·45-s + 49-s + 0.844·50-s − 4.87·52-s + 0.184·53-s − 3.17·58-s − 0.547·59-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.47·4-s − 1.20·5-s + 2.75·8-s + 9-s − 2.24·10-s − 1.96·13-s + 2.66·16-s + 1.86·18-s + 0.184·19-s − 2.98·20-s + 0.452·25-s − 3.66·26-s − 1.70·29-s + 0.891·31-s + 2.20·32-s + 2.47·36-s + 0.344·38-s − 3.32·40-s − 1.70·43-s − 1.20·45-s + 49-s + 0.844·50-s − 4.87·52-s + 0.184·53-s − 3.17·58-s − 0.547·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.483849842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483849842\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 991 | \( 1 - T \) |
| good | 2 | \( 1 - 1.86T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.20T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.96T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.184T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.70T + T^{2} \) |
| 31 | \( 1 - 0.891T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.70T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.184T + T^{2} \) |
| 59 | \( 1 + 0.547T + T^{2} \) |
| 61 | \( 1 - 0.891T + T^{2} \) |
| 67 | \( 1 - 1.47T + T^{2} \) |
| 71 | \( 1 + 0.547T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.96T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.45111628031485698689291226679, −9.652073182917745651619294999001, −8.022081760393109585123323219122, −7.24884270831555010061658610619, −6.91735689248000708967123815984, −5.55154757257054022127917544680, −4.71701612919134976367296256185, −4.13742087577338309043200699593, −3.24064205067310629735920670471, −2.08066746062596779984378033602,
2.08066746062596779984378033602, 3.24064205067310629735920670471, 4.13742087577338309043200699593, 4.71701612919134976367296256185, 5.55154757257054022127917544680, 6.91735689248000708967123815984, 7.24884270831555010061658610619, 8.022081760393109585123323219122, 9.652073182917745651619294999001, 10.45111628031485698689291226679
