L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 6·13-s + 14-s + 16-s − 6·17-s − 4·19-s + 20-s + 25-s + 6·26-s − 28-s + 6·29-s − 32-s + 6·34-s − 35-s + 10·37-s + 4·38-s − 40-s − 2·41-s + 8·43-s − 8·47-s + 49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7981995095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7981995095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.55889878951566, −12.22198589296502, −11.54668430316475, −11.18931420063681, −10.63293811459556, −10.33651768025294, −9.740710289442992, −9.335335907429005, −9.202975617114807, −8.441949528776232, −8.011959148242389, −7.633339593933075, −6.914221368412415, −6.571273469684131, −6.327673878035574, −5.643777771213277, −4.977334266952394, −4.534319154836901, −4.172133835326589, −3.201690074048980, −2.739030384023964, −2.220514552644999, −1.925781372509612, −0.9574376369112214, −0.2923436961514665,
0.2923436961514665, 0.9574376369112214, 1.925781372509612, 2.220514552644999, 2.739030384023964, 3.201690074048980, 4.172133835326589, 4.534319154836901, 4.977334266952394, 5.643777771213277, 6.327673878035574, 6.571273469684131, 6.914221368412415, 7.633339593933075, 8.011959148242389, 8.441949528776232, 9.202975617114807, 9.335335907429005, 9.740710289442992, 10.33651768025294, 10.63293811459556, 11.18931420063681, 11.54668430316475, 12.22198589296502, 12.55889878951566