L(s) = 1 | + 4·5-s + 3·7-s − 2·11-s − 5·13-s + 2·17-s + 5·19-s − 2·23-s + 11·25-s + 6·29-s + 5·31-s + 12·35-s − 3·37-s + 12·41-s − 11·43-s − 6·47-s + 2·49-s + 10·53-s − 8·55-s − 8·59-s − 2·61-s − 20·65-s − 4·67-s + 14·71-s + 10·73-s − 6·77-s − 17·79-s − 10·83-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 1.13·7-s − 0.603·11-s − 1.38·13-s + 0.485·17-s + 1.14·19-s − 0.417·23-s + 11/5·25-s + 1.11·29-s + 0.898·31-s + 2.02·35-s − 0.493·37-s + 1.87·41-s − 1.67·43-s − 0.875·47-s + 2/7·49-s + 1.37·53-s − 1.07·55-s − 1.04·59-s − 0.256·61-s − 2.48·65-s − 0.488·67-s + 1.66·71-s + 1.17·73-s − 0.683·77-s − 1.91·79-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.727683301\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.727683301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.390248478867257733562063531717, −8.385263274951441079099570192082, −7.67505998296570481063417736078, −6.80036042005189551663908081999, −5.82286714642641478643554801580, −5.14998331997603243545093226231, −4.69354605774750932201495929628, −2.93619383473491740266280363971, −2.21120386219191133993868849933, −1.23360287474869260472074540498,
1.23360287474869260472074540498, 2.21120386219191133993868849933, 2.93619383473491740266280363971, 4.69354605774750932201495929628, 5.14998331997603243545093226231, 5.82286714642641478643554801580, 6.80036042005189551663908081999, 7.67505998296570481063417736078, 8.385263274951441079099570192082, 9.390248478867257733562063531717