| L(s) = 1 | + 1.80·2-s + 0.445·3-s + 2.24·4-s + 0.801·6-s + 2.24·8-s − 0.801·9-s + 12-s + 1.80·16-s − 1.44·18-s − 1.80·19-s + 1.00·24-s − 0.801·27-s − 0.445·29-s + 1.00·32-s − 1.80·36-s + 1.80·37-s − 3.24·38-s − 1.24·43-s + 0.801·48-s + 49-s − 1.44·54-s − 0.801·57-s − 0.801·58-s + 71-s − 1.80·72-s − 1.24·73-s + 3.24·74-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.445·3-s + 2.24·4-s + 0.801·6-s + 2.24·8-s − 0.801·9-s + 12-s + 1.80·16-s − 1.44·18-s − 1.80·19-s + 1.00·24-s − 0.801·27-s − 0.445·29-s + 1.00·32-s − 1.80·36-s + 1.80·37-s − 3.24·38-s − 1.24·43-s + 0.801·48-s + 49-s − 1.44·54-s − 0.801·57-s − 0.801·58-s + 71-s − 1.80·72-s − 1.24·73-s + 3.24·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.376292310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.376292310\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.80T + T^{2} \) |
| 3 | \( 1 - 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.24T + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - 1.80T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.480105532152327597402886321389, −8.511963950730805483533931505158, −7.77319794355709658399529797980, −6.71976537633900584986348097572, −6.11764220346762438017732349981, −5.35883900664717898801667606591, −4.42320879553884568418502779802, −3.73038486337913326008770265232, −2.76957246544161024288190011392, −2.05808700313912555456813622741,
2.05808700313912555456813622741, 2.76957246544161024288190011392, 3.73038486337913326008770265232, 4.42320879553884568418502779802, 5.35883900664717898801667606591, 6.11764220346762438017732349981, 6.71976537633900584986348097572, 7.77319794355709658399529797980, 8.511963950730805483533931505158, 9.480105532152327597402886321389
