| L(s) = 1 | − 2.37·2-s − 3-s + 3.65·4-s + 5-s + 2.37·6-s + 2.87·7-s − 3.94·8-s + 9-s − 2.37·10-s − 3.37·11-s − 3.65·12-s + 5.49·13-s − 6.83·14-s − 15-s + 2.06·16-s − 5.83·17-s − 2.37·18-s + 6.36·19-s + 3.65·20-s − 2.87·21-s + 8.03·22-s − 3.64·23-s + 3.94·24-s + 25-s − 13.0·26-s − 27-s + 10.5·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.82·4-s + 0.447·5-s + 0.971·6-s + 1.08·7-s − 1.39·8-s + 0.333·9-s − 0.752·10-s − 1.01·11-s − 1.05·12-s + 1.52·13-s − 1.82·14-s − 0.258·15-s + 0.515·16-s − 1.41·17-s − 0.560·18-s + 1.46·19-s + 0.817·20-s − 0.627·21-s + 1.71·22-s − 0.760·23-s + 0.804·24-s + 0.200·25-s − 2.56·26-s − 0.192·27-s + 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 + 0.287T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 + 0.341T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.100T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 5.59T + 97T^{2} \) |
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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
−7.72569410273645785485802027425, −7.41907601030537582093158343132, −6.39299595804307465783907707093, −5.82333954272734670665364833207, −5.03509503849716089817670411356, −4.10932612980720357039450179661, −2.77706011325323578068141225815, −1.80301701181845020058930544288, −1.24772337010476572019855310198, 0,
1.24772337010476572019855310198, 1.80301701181845020058930544288, 2.77706011325323578068141225815, 4.10932612980720357039450179661, 5.03509503849716089817670411356, 5.82333954272734670665364833207, 6.39299595804307465783907707093, 7.41907601030537582093158343132, 7.72569410273645785485802027425
