L(s) = 1 | + 4-s + 5-s − 7-s − 11-s + 13-s + 16-s − 17-s + 20-s + 2·23-s + 25-s − 28-s − 35-s + 2·37-s − 41-s − 44-s + 52-s − 53-s − 55-s − 59-s − 61-s + 64-s + 65-s − 67-s − 68-s − 71-s − 73-s + 77-s + ⋯ |
L(s) = 1 | + 4-s + 5-s − 7-s − 11-s + 13-s + 16-s − 17-s + 20-s + 2·23-s + 25-s − 28-s − 35-s + 2·37-s − 41-s − 44-s + 52-s − 53-s − 55-s − 59-s − 61-s + 64-s + 65-s − 67-s − 68-s − 71-s − 73-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.529380116\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529380116\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.474181072432998060963062397277, −8.882698178099445472745560432009, −7.81731889803138309794700037110, −6.87848295485806063438474196916, −6.33196697454762282677633377689, −5.72529992830230646889720308893, −4.68142888352548852184613609608, −3.12962442138918077567621090099, −2.70825778311174001403032818834, −1.45110099876510994446914519688,
1.45110099876510994446914519688, 2.70825778311174001403032818834, 3.12962442138918077567621090099, 4.68142888352548852184613609608, 5.72529992830230646889720308893, 6.33196697454762282677633377689, 6.87848295485806063438474196916, 7.81731889803138309794700037110, 8.882698178099445472745560432009, 9.474181072432998060963062397277