| L(s) = 1 | + 0.635·2-s − 1.42·3-s − 1.59·4-s + 2.67·5-s − 0.906·6-s − 2.11·7-s − 2.28·8-s − 0.964·9-s + 1.69·10-s + 11-s + 2.27·12-s − 5.57·13-s − 1.34·14-s − 3.81·15-s + 1.73·16-s + 1.99·17-s − 0.612·18-s + 5.50·19-s − 4.26·20-s + 3.01·21-s + 0.635·22-s + 4.34·23-s + 3.26·24-s + 2.14·25-s − 3.54·26-s + 5.65·27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | + 0.449·2-s − 0.823·3-s − 0.798·4-s + 1.19·5-s − 0.370·6-s − 0.798·7-s − 0.808·8-s − 0.321·9-s + 0.537·10-s + 0.301·11-s + 0.657·12-s − 1.54·13-s − 0.358·14-s − 0.984·15-s + 0.434·16-s + 0.483·17-s − 0.144·18-s + 1.26·19-s − 0.953·20-s + 0.657·21-s + 0.135·22-s + 0.905·23-s + 0.665·24-s + 0.428·25-s − 0.694·26-s + 1.08·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178392229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178392229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 0.635T + 2T^{2} \) |
| 3 | \( 1 + 1.42T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 2.29T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 - 0.365T + 89T^{2} \) |
| 97 | \( 1 + 2.42T + 97T^{2} \) |
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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
−9.592075889684688108623447666196, −9.150472555514459223465316059925, −7.85281930723970088514847019553, −6.74519124422589321804528277791, −6.03819127338446459671286243120, −5.28015050424507553153237299758, −4.91089778833127064721049363789, −3.45918903906224108653009541520, −2.55561743539177310500128410151, −0.75134797220851315688777951453,
0.75134797220851315688777951453, 2.55561743539177310500128410151, 3.45918903906224108653009541520, 4.91089778833127064721049363789, 5.28015050424507553153237299758, 6.03819127338446459671286243120, 6.74519124422589321804528277791, 7.85281930723970088514847019553, 9.150472555514459223465316059925, 9.592075889684688108623447666196
