| L(s) = 1 | + 2.23·2-s − 3-s + 3.00·4-s − 1.81·5-s − 2.23·6-s + 1.95·7-s + 2.25·8-s + 9-s − 4.05·10-s − 3.76·11-s − 3.00·12-s − 2.59·13-s + 4.36·14-s + 1.81·15-s − 0.969·16-s − 17-s + 2.23·18-s + 4.69·19-s − 5.44·20-s − 1.95·21-s − 8.41·22-s + 8.25·23-s − 2.25·24-s − 1.71·25-s − 5.80·26-s − 27-s + 5.86·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.809·5-s − 0.913·6-s + 0.737·7-s + 0.797·8-s + 0.333·9-s − 1.28·10-s − 1.13·11-s − 0.868·12-s − 0.719·13-s + 1.16·14-s + 0.467·15-s − 0.242·16-s − 0.242·17-s + 0.527·18-s + 1.07·19-s − 1.21·20-s − 0.425·21-s − 1.79·22-s + 1.72·23-s − 0.460·24-s − 0.343·25-s − 1.13·26-s − 0.192·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 - 8.25T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 - 2.99T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 + 0.114T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 + 5.84T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 83 | \( 1 - 7.99T + 83T^{2} \) |
| 89 | \( 1 + 0.507T + 89T^{2} \) |
| 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
−7.79132543864084276693737774345, −7.13231868940260269478988618918, −6.53315520826697698618174636530, −5.36516498952917394759624089979, −4.99881629974426442090387993132, −4.64474073234848674428932174251, −3.45830314610966260846201792595, −2.92541746422055347281589197741, −1.68712259665728741985287911786, 0,
1.68712259665728741985287911786, 2.92541746422055347281589197741, 3.45830314610966260846201792595, 4.64474073234848674428932174251, 4.99881629974426442090387993132, 5.36516498952917394759624089979, 6.53315520826697698618174636530, 7.13231868940260269478988618918, 7.79132543864084276693737774345
