| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 4·13-s − 5·17-s − 19-s − 6·21-s + 2·23-s − 9·27-s − 8·29-s − 10·31-s + 6·37-s − 12·39-s + 3·41-s − 4·43-s − 4·47-s − 3·49-s + 15·51-s − 6·53-s + 3·57-s + 8·59-s + 10·61-s + 12·63-s − 67-s − 6·69-s + 12·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 1.10·13-s − 1.21·17-s − 0.229·19-s − 1.30·21-s + 0.417·23-s − 1.73·27-s − 1.48·29-s − 1.79·31-s + 0.986·37-s − 1.92·39-s + 0.468·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 2.10·51-s − 0.824·53-s + 0.397·57-s + 1.04·59-s + 1.28·61-s + 1.51·63-s − 0.122·67-s − 0.722·69-s + 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034377733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034377733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.96698820404750, −12.77647530201733, −11.98435388931819, −11.47102756571196, −11.14123191113614, −10.96860057515635, −10.68054606781472, −9.735211943946815, −9.527835527913428, −8.772076684447447, −8.382116282048037, −7.669694027394226, −7.251466115713988, −6.588397451570373, −6.380869739259928, −5.719114510489387, −5.298125638696241, −4.946811478863145, −4.218644464842928, −3.939396267536321, −3.216935931878748, −2.046368977718818, −1.816690793170787, −1.018658934582691, −0.3738280873718371,
0.3738280873718371, 1.018658934582691, 1.816690793170787, 2.046368977718818, 3.216935931878748, 3.939396267536321, 4.218644464842928, 4.946811478863145, 5.298125638696241, 5.719114510489387, 6.380869739259928, 6.588397451570373, 7.251466115713988, 7.669694027394226, 8.382116282048037, 8.772076684447447, 9.527835527913428, 9.735211943946815, 10.68054606781472, 10.96860057515635, 11.14123191113614, 11.47102756571196, 11.98435388931819, 12.77647530201733, 12.96698820404750
