| L(s) = 1 | − 2·3-s + 9-s − 4·13-s − 6·17-s + 2·19-s − 5·25-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + 12·51-s − 6·53-s − 4·57-s − 6·59-s + 8·61-s + 4·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s − 6·83-s − 12·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1.68·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s + 1.02·61-s + 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s − 0.658·83-s − 1.28·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7085048697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7085048697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.678731346329368257431074943309, −7.895164678888831912291936607752, −6.90531576344093792893651903833, −6.50364338668347614528670710226, −5.58920639964454334454555282661, −4.91171613213171916096600238802, −4.29390759056738033972615396659, −2.99977174751051762376143721143, −1.99280794987984862469967531446, −0.51891811197874657341665639572,
0.51891811197874657341665639572, 1.99280794987984862469967531446, 2.99977174751051762376143721143, 4.29390759056738033972615396659, 4.91171613213171916096600238802, 5.58920639964454334454555282661, 6.50364338668347614528670710226, 6.90531576344093792893651903833, 7.895164678888831912291936607752, 8.678731346329368257431074943309
