| L(s) = 1 | + 1.94·2-s + 2.80·4-s − 1.24·7-s + 3.51·8-s − 2.43·14-s + 4.04·16-s + 0.867·17-s − 1.80·19-s + 25-s − 3.49·28-s − 1.94·29-s + 0.445·31-s + 4.38·32-s + 1.69·34-s + 1.24·37-s − 3.51·38-s − 1.56·41-s − 0.445·43-s − 0.867·47-s + 0.554·49-s + 1.94·50-s − 0.867·53-s − 4.38·56-s − 3.80·58-s + 0.867·62-s + 4.49·64-s + 2.43·68-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.80·4-s − 1.24·7-s + 3.51·8-s − 2.43·14-s + 4.04·16-s + 0.867·17-s − 1.80·19-s + 25-s − 3.49·28-s − 1.94·29-s + 0.445·31-s + 4.38·32-s + 1.69·34-s + 1.24·37-s − 3.51·38-s − 1.56·41-s − 0.445·43-s − 0.867·47-s + 0.554·49-s + 1.94·50-s − 0.867·53-s − 4.38·56-s − 3.80·58-s + 0.867·62-s + 4.49·64-s + 2.43·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.437047371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437047371\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 - 1.94T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.24T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.867T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.94T + T^{2} \) |
| 31 | \( 1 - 0.445T + T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 + 1.56T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 + 0.867T + T^{2} \) |
| 53 | \( 1 + 0.867T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.94T + T^{2} \) |
| 89 | \( 1 - 1.56T + T^{2} \) |
| 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
−9.596633936048339420332919832640, −8.326177886742079963557185388572, −7.35445733016031836818481632200, −6.56084956662348996996110878442, −6.15941146211412081281883417396, −5.28547429221869067074877497844, −4.40661518340663455128071464105, −3.56143545541767165109979559339, −2.95586799734051401962077380209, −1.86977031176743868347098900173,
1.86977031176743868347098900173, 2.95586799734051401962077380209, 3.56143545541767165109979559339, 4.40661518340663455128071464105, 5.28547429221869067074877497844, 6.15941146211412081281883417396, 6.56084956662348996996110878442, 7.35445733016031836818481632200, 8.326177886742079963557185388572, 9.596633936048339420332919832640
