| L(s) = 1 | − 7-s − 3·9-s − 2·13-s + 2·17-s − 4·19-s + 23-s − 2·29-s − 2·37-s + 2·41-s + 4·43-s + 12·47-s + 49-s − 2·53-s + 12·59-s − 6·61-s + 3·63-s − 4·67-s + 10·73-s + 4·79-s + 9·81-s − 12·83-s + 6·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 0.371·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s − 0.768·61-s + 0.377·63-s − 0.488·67-s + 1.17·73-s + 0.450·79-s + 81-s − 1.31·83-s + 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114912752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114912752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.22907475506321, −13.78969161428472, −13.26187250093678, −12.54538581260806, −12.37785950354832, −11.70525234768979, −11.22851170506125, −10.62368443143439, −10.29181008657156, −9.550665155229039, −9.098696501425060, −8.664132047733152, −8.029675354061516, −7.528657096013062, −6.930575095115748, −6.337942051720846, −5.772741945667074, −5.348424753795458, −4.668102018131566, −3.934894170296631, −3.451755547513577, −2.545652526866848, −2.377972053205796, −1.259490038823917, −0.3659920626756407,
0.3659920626756407, 1.259490038823917, 2.377972053205796, 2.545652526866848, 3.451755547513577, 3.934894170296631, 4.668102018131566, 5.348424753795458, 5.772741945667074, 6.337942051720846, 6.930575095115748, 7.528657096013062, 8.029675354061516, 8.664132047733152, 9.098696501425060, 9.550665155229039, 10.29181008657156, 10.62368443143439, 11.22851170506125, 11.70525234768979, 12.37785950354832, 12.54538581260806, 13.26187250093678, 13.78969161428472, 14.22907475506321
