L(s) = 1 | + 2-s + (−0.5 + 0.363i)3-s + 4-s + (−0.5 + 0.363i)6-s + 8-s + (−0.190 + 0.587i)9-s + (1.30 − 0.951i)11-s + (−0.5 + 0.363i)12-s + 16-s + (−1.61 − 1.17i)17-s + (−0.190 + 0.587i)18-s + (0.618 + 1.90i)19-s + (1.30 − 0.951i)22-s + (−0.5 + 0.363i)24-s + 25-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 + 0.363i)3-s + 4-s + (−0.5 + 0.363i)6-s + 8-s + (−0.190 + 0.587i)9-s + (1.30 − 0.951i)11-s + (−0.5 + 0.363i)12-s + 16-s + (−1.61 − 1.17i)17-s + (−0.190 + 0.587i)18-s + (0.618 + 1.90i)19-s + (1.30 − 0.951i)22-s + (−0.5 + 0.363i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.970744002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.970744002\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 251 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.504493176688873358187124725386, −8.552272599108721182956403896577, −7.70659402097243230279899556184, −6.66366075442673146063393648845, −6.17586723183085153684699797146, −5.28747670265424239803874419285, −4.59661846145463986290395199498, −3.74758887455869523144583677496, −2.82380847350095772688315132651, −1.53126557955658848634939012762,
1.36425799060875297604548973382, 2.47263942200768363886577483638, 3.66792740311743405604467278088, 4.43798917687498245782455892063, 5.20229993260623766015317565465, 6.27357892238035025396747281397, 6.86868107128889817792849155034, 7.07655257765350887553915933671, 8.624285374357432262692624621824, 9.173643097047870049428652811111