| L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s + 6·11-s − 2·14-s − 16-s + 6·17-s + 19-s − 20-s + 6·22-s + 8·23-s + 25-s + 2·28-s − 4·29-s + 5·32-s + 6·34-s − 2·35-s + 4·37-s + 38-s − 3·40-s − 2·43-s − 6·44-s + 8·46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 1.06·8-s + 0.316·10-s + 1.80·11-s − 0.534·14-s − 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.223·20-s + 1.27·22-s + 1.66·23-s + 1/5·25-s + 0.377·28-s − 0.742·29-s + 0.883·32-s + 1.02·34-s − 0.338·35-s + 0.657·37-s + 0.162·38-s − 0.474·40-s − 0.304·43-s − 0.904·44-s + 1.17·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064522451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064522451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.913801051404549343095254605315, −9.336026064876748627175224561191, −8.836488963226532523388598253810, −7.44218328127442158412477604159, −6.43277558403273238107542389585, −5.81033642723526554045616411588, −4.81376783679940109514525317705, −3.73246630802271223857044732290, −3.06388291591923378327344798065, −1.15112547813740378682858966686,
1.15112547813740378682858966686, 3.06388291591923378327344798065, 3.73246630802271223857044732290, 4.81376783679940109514525317705, 5.81033642723526554045616411588, 6.43277558403273238107542389585, 7.44218328127442158412477604159, 8.836488963226532523388598253810, 9.336026064876748627175224561191, 9.913801051404549343095254605315
