| L(s) = 1 | + 4·7-s + 6·13-s − 2·17-s − 4·19-s + 8·23-s + 6·29-s + 6·37-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s + 10·53-s + 6·61-s − 4·67-s + 14·73-s − 16·79-s − 12·83-s − 2·89-s + 24·91-s − 2·97-s + 14·101-s + 4·103-s − 4·107-s − 10·109-s + 6·113-s − 8·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1.11·29-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.768·61-s − 0.488·67-s + 1.63·73-s − 1.80·79-s − 1.31·83-s − 0.211·89-s + 2.51·91-s − 0.203·97-s + 1.39·101-s + 0.394·103-s − 0.386·107-s − 0.957·109-s + 0.564·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.571475339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571475339\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.505393237819314779609790798270, −8.089044200754267907073680217618, −6.98151774392443190375452463543, −6.41974214419273632716207435396, −5.43149038568839214981741928156, −4.73683207096237708618937222163, −4.04585704062967792203675398162, −2.99099948728210108271675714687, −1.84201706295129075260047249403, −1.03255515630969299327928050189,
1.03255515630969299327928050189, 1.84201706295129075260047249403, 2.99099948728210108271675714687, 4.04585704062967792203675398162, 4.73683207096237708618937222163, 5.43149038568839214981741928156, 6.41974214419273632716207435396, 6.98151774392443190375452463543, 8.089044200754267907073680217618, 8.505393237819314779609790798270
