| L(s) = 1 | − 2·4-s + 4.63·7-s − 5.69·13-s + 4·16-s + 4.28·19-s − 5·25-s − 9.26·28-s − 8.64·31-s + 11.9·37-s − 13.0·43-s + 14.4·49-s + 11.3·52-s − 4.21·61-s − 8·64-s − 1.77·67-s − 16.2·73-s − 8.56·76-s − 3.39·79-s − 26.3·91-s − 17.5·97-s + 10·100-s + 2.88·103-s + 16.7·109-s + 18.5·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.75·7-s − 1.57·13-s + 16-s + 0.982·19-s − 25-s − 1.75·28-s − 1.55·31-s + 1.96·37-s − 1.99·43-s + 2.06·49-s + 1.57·52-s − 0.540·61-s − 64-s − 0.216·67-s − 1.90·73-s − 0.982·76-s − 0.381·79-s − 2.76·91-s − 1.77·97-s + 100-s + 0.284·103-s + 1.60·109-s + 1.75·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.63T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 3.39T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.66346692297959276880566685506, −7.17746938006072002728213330433, −5.80301203246656688854737903269, −5.30432906245248856915194150942, −4.69190740101543008274899999648, −4.22779291434716124932379545367, −3.18458874315738660889833243891, −2.10436212484739088754172804614, −1.28486526853184290613784953983, 0,
1.28486526853184290613784953983, 2.10436212484739088754172804614, 3.18458874315738660889833243891, 4.22779291434716124932379545367, 4.69190740101543008274899999648, 5.30432906245248856915194150942, 5.80301203246656688854737903269, 7.17746938006072002728213330433, 7.66346692297959276880566685506
