| L(s) = 1 | − 2-s + 1.56·3-s + 4-s − 1.28·5-s − 1.56·6-s − 7-s − 8-s − 0.563·9-s + 1.28·10-s − 0.685·11-s + 1.56·12-s − 1.48·13-s + 14-s − 2.00·15-s + 16-s − 5.29·17-s + 0.563·18-s − 3.49·19-s − 1.28·20-s − 1.56·21-s + 0.685·22-s + 4.10·23-s − 1.56·24-s − 3.34·25-s + 1.48·26-s − 5.56·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.901·3-s + 0.5·4-s − 0.575·5-s − 0.637·6-s − 0.377·7-s − 0.353·8-s − 0.187·9-s + 0.407·10-s − 0.206·11-s + 0.450·12-s − 0.411·13-s + 0.267·14-s − 0.518·15-s + 0.250·16-s − 1.28·17-s + 0.132·18-s − 0.802·19-s − 0.287·20-s − 0.340·21-s + 0.146·22-s + 0.856·23-s − 0.318·24-s − 0.668·25-s + 0.290·26-s − 1.07·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9603886715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9603886715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 11 | \( 1 + 0.685T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 - 0.0567T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 9.13T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 1.36T + 89T^{2} \) |
| 97 | \( 1 + 6.23T + 97T^{2} \) |
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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
−8.204479867201185958556686939151, −7.47944128939484266678311801729, −6.97651937368084258904121373744, −6.13128572956440713938825486824, −5.25149971697968853419124606724, −4.14878412328748728670215207651, −3.56691079841657386565999514315, −2.52156326911436302926813179334, −2.13341776086793481722700618568, −0.51152000154552941138218756977,
0.51152000154552941138218756977, 2.13341776086793481722700618568, 2.52156326911436302926813179334, 3.56691079841657386565999514315, 4.14878412328748728670215207651, 5.25149971697968853419124606724, 6.13128572956440713938825486824, 6.97651937368084258904121373744, 7.47944128939484266678311801729, 8.204479867201185958556686939151
