L(s) = 1 | + 2·5-s − 6·13-s − 17-s − 8·19-s − 2·23-s − 25-s + 6·29-s − 8·31-s − 4·37-s + 10·41-s − 8·43-s + 4·47-s + 10·53-s + 14·59-s + 6·61-s − 12·65-s + 4·67-s + 6·71-s − 10·73-s + 4·79-s + 6·83-s − 2·85-s + 6·89-s − 16·95-s + 18·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.66·13-s − 0.242·17-s − 1.83·19-s − 0.417·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.657·37-s + 1.56·41-s − 1.21·43-s + 0.583·47-s + 1.37·53-s + 1.82·59-s + 0.768·61-s − 1.48·65-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 0.450·79-s + 0.658·83-s − 0.216·85-s + 0.635·89-s − 1.64·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.86364777906926, −13.16811309137626, −12.94169324408101, −12.43682106242530, −11.91113822202791, −11.48720805449091, −10.66520032228657, −10.40122627092515, −9.963343796986512, −9.481961122946198, −8.916214447770285, −8.513404613562336, −7.893690901609555, −7.250349739506515, −6.851752354235263, −6.310555372262677, −5.756883477566619, −5.236053543717768, −4.736797077373348, −4.117085199647060, −3.590095345841213, −2.579348158341866, −2.207898046898368, −1.923623327984015, −0.7795190798636442, 0,
0.7795190798636442, 1.923623327984015, 2.207898046898368, 2.579348158341866, 3.590095345841213, 4.117085199647060, 4.736797077373348, 5.236053543717768, 5.756883477566619, 6.310555372262677, 6.851752354235263, 7.250349739506515, 7.893690901609555, 8.513404613562336, 8.916214447770285, 9.481961122946198, 9.963343796986512, 10.40122627092515, 10.66520032228657, 11.48720805449091, 11.91113822202791, 12.43682106242530, 12.94169324408101, 13.16811309137626, 13.86364777906926