| L(s) = 1 | + 2·2-s + 3·4-s − 5-s + 4·8-s + 9-s − 2·10-s − 11-s − 13-s + 5·16-s + 2·18-s − 3·20-s − 2·22-s − 23-s − 2·26-s − 31-s + 6·32-s + 3·36-s − 37-s − 4·40-s + 2·41-s − 3·44-s − 45-s − 2·46-s + 49-s − 3·52-s − 53-s + 55-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 5-s + 4·8-s + 9-s − 2·10-s − 11-s − 13-s + 5·16-s + 2·18-s − 3·20-s − 2·22-s − 23-s − 2·26-s − 31-s + 6·32-s + 3·36-s − 37-s − 4·40-s + 2·41-s − 3·44-s − 45-s − 2·46-s + 49-s − 3·52-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.453886081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.453886081\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1999 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.620704835491361841755602294288, −7.83728996943413806100891005429, −7.63943538744656565721528388520, −6.93468207523552203207097409479, −5.93019216613148816800310910674, −5.11502300663308418273011260303, −4.38723461222590729767556671261, −3.82991464292252792437144147835, −2.83517005917974515641863664218, −1.88236955051199998682652391443,
1.88236955051199998682652391443, 2.83517005917974515641863664218, 3.82991464292252792437144147835, 4.38723461222590729767556671261, 5.11502300663308418273011260303, 5.93019216613148816800310910674, 6.93468207523552203207097409479, 7.63943538744656565721528388520, 7.83728996943413806100891005429, 9.620704835491361841755602294288
