| L(s) = 1 | + 3-s + 4·5-s + 2·7-s + 9-s − 2·11-s + 4·15-s + 6·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s + 27-s + 6·29-s − 10·31-s − 2·33-s + 8·35-s + 4·37-s + 4·43-s + 4·45-s + 2·47-s − 3·49-s + 6·51-s − 10·53-s − 8·55-s + 2·57-s + 14·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.03·15-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.348·33-s + 1.35·35-s + 0.657·37-s + 0.609·43-s + 0.596·45-s + 0.291·47-s − 3/7·49-s + 0.840·51-s − 1.37·53-s − 1.07·55-s + 0.264·57-s + 1.82·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.871222338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.871222338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.308283855337554219831439565870, −7.907105534740594504521415808977, −7.01651452318300055958386162526, −6.03204597295832233627215963554, −5.51499788544589681968042938827, −4.89076481154533937825130256543, −3.73022230767307202515891568855, −2.70761732532164436849768211909, −2.00569056698838497774876607443, −1.22386450874973274875871395128,
1.22386450874973274875871395128, 2.00569056698838497774876607443, 2.70761732532164436849768211909, 3.73022230767307202515891568855, 4.89076481154533937825130256543, 5.51499788544589681968042938827, 6.03204597295832233627215963554, 7.01651452318300055958386162526, 7.907105534740594504521415808977, 8.308283855337554219831439565870
