L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 4·11-s + 13-s + 15-s + 17-s + 4·19-s − 4·21-s + 4·23-s + 25-s − 27-s − 6·29-s − 4·31-s − 4·33-s − 4·35-s + 2·37-s − 39-s − 6·41-s + 12·43-s − 45-s + 8·47-s + 9·49-s − 51-s − 10·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.160·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.140·51-s − 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.721572544\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.721572544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.33182406203617, −14.69488867126067, −14.26523867738871, −13.91823898878472, −13.04587996023631, −12.49046468389251, −11.91400636223830, −11.48512274781114, −11.03926921291581, −10.78400764045921, −9.788615285001783, −9.222531999157888, −8.761269450194739, −8.006356303407280, −7.489460444593242, −7.063462798377110, −6.296320137959948, −5.506402155367709, −5.186682892713201, −4.361286836630124, −3.939217889783601, −3.180897867686470, −2.049240157783777, −1.352556207405079, −0.7598668528030454,
0.7598668528030454, 1.352556207405079, 2.049240157783777, 3.180897867686470, 3.939217889783601, 4.361286836630124, 5.186682892713201, 5.506402155367709, 6.296320137959948, 7.063462798377110, 7.489460444593242, 8.006356303407280, 8.761269450194739, 9.222531999157888, 9.788615285001783, 10.78400764045921, 11.03926921291581, 11.48512274781114, 11.91400636223830, 12.49046468389251, 13.04587996023631, 13.91823898878472, 14.26523867738871, 14.69488867126067, 15.33182406203617