| L(s) = 1 | + 2.21·2-s − 2.47·3-s + 2.89·4-s + 2.12·5-s − 5.47·6-s + 1.98·8-s + 3.12·9-s + 4.69·10-s + 4.78·11-s − 7.17·12-s − 13-s − 5.25·15-s − 1.39·16-s + 3.77·17-s + 6.91·18-s + 3.56·19-s + 6.15·20-s + 10.5·22-s + 4.47·23-s − 4.92·24-s − 0.493·25-s − 2.21·26-s − 0.303·27-s − 5.90·29-s − 11.6·30-s + 3.77·31-s − 7.06·32-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 1.42·3-s + 1.44·4-s + 0.949·5-s − 2.23·6-s + 0.703·8-s + 1.04·9-s + 1.48·10-s + 1.44·11-s − 2.07·12-s − 0.277·13-s − 1.35·15-s − 0.348·16-s + 0.916·17-s + 1.62·18-s + 0.818·19-s + 1.37·20-s + 2.25·22-s + 0.932·23-s − 1.00·24-s − 0.0987·25-s − 0.434·26-s − 0.0584·27-s − 1.09·29-s − 2.12·30-s + 0.677·31-s − 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.708447937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708447937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 + 2.89T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 5.17T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 3.66T + 89T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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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
−11.02249841820461264511946933667, −9.928884774555821449613823913388, −9.160740331945007267013922121026, −7.31094096748990328819136361717, −6.43648383391597638332030993584, −5.86029333702228367881185780023, −5.23663060488091602639978656626, −4.34012735425512558921969304698, −3.12425163027823506176197906099, −1.44278281573533867190536644099,
1.44278281573533867190536644099, 3.12425163027823506176197906099, 4.34012735425512558921969304698, 5.23663060488091602639978656626, 5.86029333702228367881185780023, 6.43648383391597638332030993584, 7.31094096748990328819136361717, 9.160740331945007267013922121026, 9.928884774555821449613823913388, 11.02249841820461264511946933667
