L(s) = 1 | + 5-s − 2·7-s − 2·13-s + 6·17-s + 6·23-s + 25-s − 4·29-s − 2·35-s − 10·37-s − 8·41-s − 2·43-s − 2·47-s − 3·49-s − 2·53-s + 14·61-s − 2·65-s + 4·67-s − 12·71-s + 6·73-s − 8·79-s + 2·83-s + 6·85-s − 8·89-s + 4·91-s + 6·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s − 0.742·29-s − 0.338·35-s − 1.64·37-s − 1.24·41-s − 0.304·43-s − 0.291·47-s − 3/7·49-s − 0.274·53-s + 1.79·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.900·79-s + 0.219·83-s + 0.650·85-s − 0.847·89-s + 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.418875083\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418875083\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.92317665094899, −12.28454821426484, −12.07503733357592, −11.43911507288035, −10.96871939052627, −10.33777050690007, −10.02231130026400, −9.654827237564252, −9.226548602201995, −8.576524469242836, −8.293270987762998, −7.473533577132942, −7.179065175592622, −6.691958358770027, −6.227715864620002, −5.530678844385170, −5.196722115925360, −4.867207114539798, −3.879131325563981, −3.511763857703667, −3.002046042125418, −2.503487223742229, −1.667598727255265, −1.247508300860976, −0.3244235800575461,
0.3244235800575461, 1.247508300860976, 1.667598727255265, 2.503487223742229, 3.002046042125418, 3.511763857703667, 3.879131325563981, 4.867207114539798, 5.196722115925360, 5.530678844385170, 6.227715864620002, 6.691958358770027, 7.179065175592622, 7.473533577132942, 8.293270987762998, 8.576524469242836, 9.226548602201995, 9.654827237564252, 10.02231130026400, 10.33777050690007, 10.96871939052627, 11.43911507288035, 12.07503733357592, 12.28454821426484, 12.92317665094899