| L(s) = 1 | − 5-s + 2·7-s − 3·9-s − 2·11-s − 6·13-s + 2·17-s + 2·19-s + 6·23-s + 25-s − 29-s + 6·31-s − 2·35-s − 2·37-s + 10·41-s + 8·43-s + 3·45-s + 4·47-s − 3·49-s + 10·53-s + 2·55-s − 8·59-s + 10·61-s − 6·63-s + 6·65-s − 2·67-s − 4·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s − 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.185·29-s + 1.07·31-s − 0.338·35-s − 0.328·37-s + 1.56·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 1.04·59-s + 1.28·61-s − 0.755·63-s + 0.744·65-s − 0.244·67-s − 0.474·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394440487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394440487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.989893394223599149812160498767, −8.030109914217494010498465613687, −7.65510238158751018228128534275, −6.85175410143401188714284594587, −5.59085879535762716730664686890, −5.12036148920864175599945732516, −4.29495188407902907687575262766, −2.99743557149906161555251361231, −2.40015931630083824812790117819, −0.74599768200162322439789091075,
0.74599768200162322439789091075, 2.40015931630083824812790117819, 2.99743557149906161555251361231, 4.29495188407902907687575262766, 5.12036148920864175599945732516, 5.59085879535762716730664686890, 6.85175410143401188714284594587, 7.65510238158751018228128534275, 8.030109914217494010498465613687, 8.989893394223599149812160498767
