L(s) = 1 | − 7-s − 3·9-s − 2·11-s − 4·13-s + 6·17-s + 8·19-s + 23-s + 10·29-s − 10·31-s − 8·37-s − 2·41-s + 12·47-s + 49-s + 4·53-s − 14·59-s − 2·61-s + 3·63-s − 4·67-s − 8·71-s + 2·77-s − 6·79-s + 9·81-s − 12·83-s + 10·89-s + 4·91-s + 2·97-s + 6·99-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.83·19-s + 0.208·23-s + 1.85·29-s − 1.79·31-s − 1.31·37-s − 0.312·41-s + 1.75·47-s + 1/7·49-s + 0.549·53-s − 1.82·59-s − 0.256·61-s + 0.377·63-s − 0.488·67-s − 0.949·71-s + 0.227·77-s − 0.675·79-s + 81-s − 1.31·83-s + 1.05·89-s + 0.419·91-s + 0.203·97-s + 0.603·99-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.46034609882210, −13.92892382745838, −13.72089997366212, −12.90937802236813, −12.31425309366898, −11.91600399824963, −11.79927967898195, −10.71740452446533, −10.50429114880943, −9.931085558955837, −9.324721681816243, −8.990427217640466, −8.248811129395021, −7.665057966470373, −7.327987548437171, −6.799684688279551, −5.827971372325279, −5.517559241895964, −5.153421058330104, −4.423501193786507, −3.444750007552784, −3.048590719456757, −2.678617964323737, −1.685066574153302, −0.8302780139144496, 0,
0.8302780139144496, 1.685066574153302, 2.678617964323737, 3.048590719456757, 3.444750007552784, 4.423501193786507, 5.153421058330104, 5.517559241895964, 5.827971372325279, 6.799684688279551, 7.327987548437171, 7.665057966470373, 8.248811129395021, 8.990427217640466, 9.324721681816243, 9.931085558955837, 10.50429114880943, 10.71740452446533, 11.79927967898195, 11.91600399824963, 12.31425309366898, 12.90937802236813, 13.72089997366212, 13.92892382745838, 14.46034609882210