| L(s) = 1 | + 7·3-s − 7·7-s + 22·9-s − 7·11-s + 3·13-s + 61·17-s + 48·19-s − 49·21-s + 58·23-s − 35·27-s + 219·29-s + 298·31-s − 49·33-s − 170·37-s + 21·39-s + 50·41-s + 484·43-s + 131·47-s + 49·49-s + 427·51-s + 210·53-s + 336·57-s − 782·59-s + 488·61-s − 154·63-s + 494·67-s + 406·69-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.377·7-s + 0.814·9-s − 0.191·11-s + 0.0640·13-s + 0.870·17-s + 0.579·19-s − 0.509·21-s + 0.525·23-s − 0.249·27-s + 1.40·29-s + 1.72·31-s − 0.258·33-s − 0.755·37-s + 0.0862·39-s + 0.190·41-s + 1.71·43-s + 0.406·47-s + 1/7·49-s + 1.17·51-s + 0.544·53-s + 0.780·57-s − 1.72·59-s + 1.02·61-s − 0.307·63-s + 0.900·67-s + 0.708·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.347455953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.347455953\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 - 3 T + p^{3} T^{2} \) |
| 17 | \( 1 - 61 T + p^{3} T^{2} \) |
| 19 | \( 1 - 48 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 219 T + p^{3} T^{2} \) |
| 31 | \( 1 - 298 T + p^{3} T^{2} \) |
| 37 | \( 1 + 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 50 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 131 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 + 782 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 494 T + p^{3} T^{2} \) |
| 71 | \( 1 + 240 T + p^{3} T^{2} \) |
| 73 | \( 1 - 58 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1065 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 608 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1339 T + p^{3} T^{2} \) |
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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
−9.865113457964303170453193999923, −9.152370805964178258474862493077, −8.325996116236518303945787234467, −7.67183427871722569105887560384, −6.69867259189124770020212974285, −5.53468466864200972996864492897, −4.29774960242756491319204990451, −3.19549368866538100015163340093, −2.57242081031577782729882998591, −1.03951510157037679992252829965,
1.03951510157037679992252829965, 2.57242081031577782729882998591, 3.19549368866538100015163340093, 4.29774960242756491319204990451, 5.53468466864200972996864492897, 6.69867259189124770020212974285, 7.67183427871722569105887560384, 8.325996116236518303945787234467, 9.152370805964178258474862493077, 9.865113457964303170453193999923
