| L(s) = 1 | − 1.75·2-s + 2.94·3-s + 1.09·4-s − 3.30·5-s − 5.18·6-s − 3.55·7-s + 1.59·8-s + 5.70·9-s + 5.82·10-s − 1.01·11-s + 3.22·12-s − 13-s + 6.26·14-s − 9.76·15-s − 4.99·16-s − 6.98·17-s − 10.0·18-s + 2.60·19-s − 3.62·20-s − 10.4·21-s + 1.78·22-s − 8.89·23-s + 4.69·24-s + 5.95·25-s + 1.75·26-s + 7.96·27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 1.70·3-s + 0.547·4-s − 1.48·5-s − 2.11·6-s − 1.34·7-s + 0.562·8-s + 1.90·9-s + 1.84·10-s − 0.306·11-s + 0.932·12-s − 0.277·13-s + 1.67·14-s − 2.52·15-s − 1.24·16-s − 1.69·17-s − 2.36·18-s + 0.596·19-s − 0.810·20-s − 2.29·21-s + 0.381·22-s − 1.85·23-s + 0.958·24-s + 1.19·25-s + 0.345·26-s + 1.53·27-s − 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.75T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 + 0.484T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 - 7.46T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.39T + 67T^{2} \) |
| 71 | \( 1 + 5.94T + 71T^{2} \) |
| 73 | \( 1 - 9.48T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−10.30606845939151991649885098953, −9.669197978646569331037290655936, −8.808152351697551900694411360464, −8.275641285833424983803978837038, −7.48506444968853689207321049834, −6.79627740277021879355454576150, −4.32818034571112468288800247797, −3.53484852311102144477275489243, −2.31023457792955076016158407074, 0,
2.31023457792955076016158407074, 3.53484852311102144477275489243, 4.32818034571112468288800247797, 6.79627740277021879355454576150, 7.48506444968853689207321049834, 8.275641285833424983803978837038, 8.808152351697551900694411360464, 9.669197978646569331037290655936, 10.30606845939151991649885098953
