| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 2·11-s − 12-s + 16-s − 18-s − 2·22-s + 24-s − 25-s − 27-s − 32-s − 2·33-s + 36-s + 2·44-s + 2·47-s − 48-s + 49-s + 50-s + 54-s − 2·61-s + 64-s + 2·66-s − 72-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 2·11-s − 12-s + 16-s − 18-s − 2·22-s + 24-s − 25-s − 27-s − 32-s − 2·33-s + 36-s + 2·44-s + 2·47-s − 48-s + 49-s + 50-s + 54-s − 2·61-s + 64-s + 2·66-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6123726098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6123726098\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.308221792464370107966330618651, −8.835102660578164800194712866030, −7.63600232900437909422550114902, −7.04403072591249159288347402236, −6.23222278875595057811754264983, −5.78275465349474400265218232817, −4.42079754777422601426895781678, −3.57704250543629499650426375885, −1.98189895775006332408662433363, −0.996403235650219158516985750825,
0.996403235650219158516985750825, 1.98189895775006332408662433363, 3.57704250543629499650426375885, 4.42079754777422601426895781678, 5.78275465349474400265218232817, 6.23222278875595057811754264983, 7.04403072591249159288347402236, 7.63600232900437909422550114902, 8.835102660578164800194712866030, 9.308221792464370107966330618651
