| L(s) = 1 | + 1.97·2-s + 2.65·3-s + 1.91·4-s − 5-s + 5.24·6-s + 1.06·7-s − 0.173·8-s + 4.04·9-s − 1.97·10-s + 11-s + 5.07·12-s + 0.939·13-s + 2.10·14-s − 2.65·15-s − 4.16·16-s + 3.65·17-s + 7.99·18-s − 19-s − 1.91·20-s + 2.83·21-s + 1.97·22-s + 5.35·23-s − 0.461·24-s + 25-s + 1.85·26-s + 2.77·27-s + 2.03·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 1.53·3-s + 0.956·4-s − 0.447·5-s + 2.14·6-s + 0.403·7-s − 0.0614·8-s + 1.34·9-s − 0.625·10-s + 0.301·11-s + 1.46·12-s + 0.260·13-s + 0.563·14-s − 0.685·15-s − 1.04·16-s + 0.887·17-s + 1.88·18-s − 0.229·19-s − 0.427·20-s + 0.617·21-s + 0.421·22-s + 1.11·23-s − 0.0941·24-s + 0.200·25-s + 0.364·26-s + 0.533·27-s + 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.118618950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.118618950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 - 2.65T + 3T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 13 | \( 1 - 0.939T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 2.71T + 31T^{2} \) |
| 37 | \( 1 - 0.932T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 4.30T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 - 2.13T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 - 7.32T + 71T^{2} \) |
| 73 | \( 1 - 1.14T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 8.77T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.702849676393408216659470146192, −8.998516713282529631693217601854, −8.203020646760703938362325125296, −7.45334247104831911003816357591, −6.50580697830643175347111262684, −5.32321021309886494535961641219, −4.44856510659346713751510685672, −3.52463813550576053830771804103, −3.07104096336660036267224358907, −1.78669165550340696671943807640,
1.78669165550340696671943807640, 3.07104096336660036267224358907, 3.52463813550576053830771804103, 4.44856510659346713751510685672, 5.32321021309886494535961641219, 6.50580697830643175347111262684, 7.45334247104831911003816357591, 8.203020646760703938362325125296, 8.998516713282529631693217601854, 9.702849676393408216659470146192
