| L(s) = 1 | + 5-s − 3·7-s + 2·11-s + 5·13-s − 8·17-s − 19-s + 6·23-s + 25-s − 2·29-s − 3·35-s − 5·37-s − 10·41-s − 4·43-s + 4·47-s + 2·49-s + 2·53-s + 2·55-s + 8·59-s − 7·61-s + 5·65-s + 9·67-s + 2·71-s − 5·73-s − 6·77-s − 3·79-s − 6·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.603·11-s + 1.38·13-s − 1.94·17-s − 0.229·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s − 0.507·35-s − 0.821·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 2/7·49-s + 0.274·53-s + 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.620·65-s + 1.09·67-s + 0.237·71-s − 0.585·73-s − 0.683·77-s − 0.337·79-s − 0.658·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.01519239662859799823423413506, −6.77007801705803662555448513376, −6.21925386202534691881698900223, −5.47063726054456732355373835481, −4.56519680808379438150600774763, −3.76215660304406992002251700250, −3.17093417750280844721637860838, −2.19720414736728443843018214929, −1.27847403078412882027024062749, 0,
1.27847403078412882027024062749, 2.19720414736728443843018214929, 3.17093417750280844721637860838, 3.76215660304406992002251700250, 4.56519680808379438150600774763, 5.47063726054456732355373835481, 6.21925386202534691881698900223, 6.77007801705803662555448513376, 7.01519239662859799823423413506
