| L(s) = 1 | + 0.732·3-s + 5-s − 2·7-s − 2.46·9-s − 3.46·11-s + 0.732·13-s + 0.732·15-s + 3.46·17-s − 19-s − 1.46·21-s − 3.46·23-s + 25-s − 4·27-s − 3.46·29-s − 5.46·31-s − 2.53·33-s − 2·35-s + 3.26·37-s + 0.535·39-s − 6·41-s − 8.92·43-s − 2.46·45-s + 0.928·47-s − 3·49-s + 2.53·51-s − 7.26·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.447·5-s − 0.755·7-s − 0.821·9-s − 1.04·11-s + 0.203·13-s + 0.189·15-s + 0.840·17-s − 0.229·19-s − 0.319·21-s − 0.722·23-s + 0.200·25-s − 0.769·27-s − 0.643·29-s − 0.981·31-s − 0.441·33-s − 0.338·35-s + 0.537·37-s + 0.0858·39-s − 0.937·41-s − 1.36·43-s − 0.367·45-s + 0.135·47-s − 0.428·49-s + 0.355·51-s − 0.998·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−9.131492239069850516877154401619, −8.236698311914008664717230394914, −7.65297966437495532642683286587, −6.52911533516102420844842400796, −5.77062955146184680594812156109, −5.08084977293120133769417440291, −3.63556900134341021233384687215, −2.96373216224476934882459709097, −1.92361700076206183489253495483, 0,
1.92361700076206183489253495483, 2.96373216224476934882459709097, 3.63556900134341021233384687215, 5.08084977293120133769417440291, 5.77062955146184680594812156109, 6.52911533516102420844842400796, 7.65297966437495532642683286587, 8.236698311914008664717230394914, 9.131492239069850516877154401619
