L(s) = 1 | + 3·7-s − 3·9-s + 3·11-s − 17-s + 4·19-s + 2·23-s − 7·29-s − 5·31-s + 6·37-s − 4·41-s − 6·43-s + 13·47-s + 2·49-s − 9·53-s + 5·59-s − 13·61-s − 9·63-s − 5·67-s + 2·73-s + 9·77-s − 14·79-s + 9·81-s − 9·83-s − 4·89-s + 2·97-s − 9·99-s + 101-s + ⋯ |
L(s) = 1 | + 1.13·7-s − 9-s + 0.904·11-s − 0.242·17-s + 0.917·19-s + 0.417·23-s − 1.29·29-s − 0.898·31-s + 0.986·37-s − 0.624·41-s − 0.914·43-s + 1.89·47-s + 2/7·49-s − 1.23·53-s + 0.650·59-s − 1.66·61-s − 1.13·63-s − 0.610·67-s + 0.234·73-s + 1.02·77-s − 1.57·79-s + 81-s − 0.987·83-s − 0.423·89-s + 0.203·97-s − 0.904·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.199685340\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.199685340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.82259349055765, −12.15212991222867, −11.72592549835091, −11.43419068188241, −11.07658123041487, −10.66672526785978, −9.996327845988161, −9.314190073631541, −9.140559433044478, −8.641571404997044, −8.130837482819576, −7.604907883877531, −7.285727925784732, −6.659352475679452, −6.045567478043788, −5.562817818250674, −5.260893165965113, −4.556533221541466, −4.171494526706458, −3.457724655352670, −3.039566206432052, −2.339173886735383, −1.645751317217160, −1.304456371942477, −0.3986877738544503,
0.3986877738544503, 1.304456371942477, 1.645751317217160, 2.339173886735383, 3.039566206432052, 3.457724655352670, 4.171494526706458, 4.556533221541466, 5.260893165965113, 5.562817818250674, 6.045567478043788, 6.659352475679452, 7.285727925784732, 7.604907883877531, 8.130837482819576, 8.641571404997044, 9.140559433044478, 9.314190073631541, 9.996327845988161, 10.66672526785978, 11.07658123041487, 11.43419068188241, 11.72592549835091, 12.15212991222867, 12.82259349055765