L(s) = 1 | − 7·13-s + 7·19-s − 5·25-s + 7·31-s − 37-s − 5·43-s + 14·61-s − 11·67-s − 7·73-s + 13·79-s + 14·97-s + 7·103-s + 17·109-s + ⋯ |
L(s) = 1 | − 1.94·13-s + 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s − 0.762·43-s + 1.79·61-s − 1.34·67-s − 0.819·73-s + 1.46·79-s + 1.42·97-s + 0.689·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.599591298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.599591298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.73594011226772717557429578921, −7.38837049753910494835648660850, −6.64356021456249165343568157928, −5.74202906155295297295597884206, −5.06671097741570885388825087277, −4.51783710729529372415064306443, −3.46966062525893120155742621808, −2.72502754714260381206626087168, −1.89236012616310691619963980542, −0.62873387229788762149874796515,
0.62873387229788762149874796515, 1.89236012616310691619963980542, 2.72502754714260381206626087168, 3.46966062525893120155742621808, 4.51783710729529372415064306443, 5.06671097741570885388825087277, 5.74202906155295297295597884206, 6.64356021456249165343568157928, 7.38837049753910494835648660850, 7.73594011226772717557429578921