| L(s) = 1 | + 2-s − 3.22·3-s + 4-s − 1.74·5-s − 3.22·6-s + 8-s + 7.37·9-s − 1.74·10-s − 1.73·11-s − 3.22·12-s + 3.15·13-s + 5.60·15-s + 16-s − 2.16·17-s + 7.37·18-s − 1.23·19-s − 1.74·20-s − 1.73·22-s + 0.783·23-s − 3.22·24-s − 1.96·25-s + 3.15·26-s − 14.0·27-s + 0.362·29-s + 5.60·30-s − 6.04·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.85·3-s + 0.5·4-s − 0.779·5-s − 1.31·6-s + 0.353·8-s + 2.45·9-s − 0.550·10-s − 0.522·11-s − 0.929·12-s + 0.876·13-s + 1.44·15-s + 0.250·16-s − 0.525·17-s + 1.73·18-s − 0.282·19-s − 0.389·20-s − 0.369·22-s + 0.163·23-s − 0.657·24-s − 0.393·25-s + 0.619·26-s − 2.70·27-s + 0.0673·29-s + 1.02·30-s − 1.08·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 0.783T + 23T^{2} \) |
| 29 | \( 1 - 0.362T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 + 0.455T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 0.302T + 67T^{2} \) |
| 71 | \( 1 - 5.60T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 0.308T + 97T^{2} \) |
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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
−7.38628306190827826973652775257, −6.54144037947551254878952286029, −6.03068645227049966480457351028, −5.48123700579182800783010294357, −4.70244847807243354544424259488, −4.17974111471726520768755893482, −3.49629926425324127623341853388, −2.16437863701045841407462308394, −1.03919127746105266637002266459, 0,
1.03919127746105266637002266459, 2.16437863701045841407462308394, 3.49629926425324127623341853388, 4.17974111471726520768755893482, 4.70244847807243354544424259488, 5.48123700579182800783010294357, 6.03068645227049966480457351028, 6.54144037947551254878952286029, 7.38628306190827826973652775257
