L(s) = 1 | + 1.53·2-s + 3.17·3-s + 0.369·4-s + 4.87·6-s − 1.70·7-s − 2.51·8-s + 7.04·9-s − 2.53·11-s + 1.17·12-s − 13-s − 2.63·14-s − 4.60·16-s + 0.921·17-s + 10.8·18-s − 0.539·19-s − 5.41·21-s − 3.90·22-s + 2.82·23-s − 7.95·24-s − 1.53·26-s + 12.8·27-s − 0.630·28-s − 5.12·29-s + 0.879·31-s − 2.06·32-s − 8.04·33-s + 1.41·34-s + ⋯ |
L(s) = 1 | + 1.08·2-s + 1.83·3-s + 0.184·4-s + 1.99·6-s − 0.646·7-s − 0.887·8-s + 2.34·9-s − 0.765·11-s + 0.337·12-s − 0.277·13-s − 0.703·14-s − 1.15·16-s + 0.223·17-s + 2.55·18-s − 0.123·19-s − 1.18·21-s − 0.833·22-s + 0.590·23-s − 1.62·24-s − 0.301·26-s + 2.47·27-s − 0.119·28-s − 0.952·29-s + 0.157·31-s − 0.364·32-s − 1.40·33-s + 0.243·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.142664030\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.142664030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 17 | \( 1 - 0.921T + 17T^{2} \) |
| 19 | \( 1 + 0.539T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 0.879T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 + 4.72T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 - 0.496T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−12.12941098663550587673867455130, −10.51568653098764028919599298345, −9.490992284725866598428464227527, −8.885913318848338799350511686252, −7.82987191263335179474453138131, −6.85440613815811779037615836976, −5.40827252842327508994339294506, −4.16586318715189151702457031697, −3.28343035023407878853429763755, −2.43403353966411695770152814782,
2.43403353966411695770152814782, 3.28343035023407878853429763755, 4.16586318715189151702457031697, 5.40827252842327508994339294506, 6.85440613815811779037615836976, 7.82987191263335179474453138131, 8.885913318848338799350511686252, 9.490992284725866598428464227527, 10.51568653098764028919599298345, 12.12941098663550587673867455130