| L(s) = 1 | + 0.291·3-s − 0.918·5-s − 2.91·9-s + 4.23·11-s + 2.07·13-s − 0.267·15-s − 17-s − 6.92·19-s + 1.49·23-s − 4.15·25-s − 1.72·27-s + 10.0·29-s − 1.01·31-s + 1.23·33-s − 6.04·37-s + 0.605·39-s + 3.68·41-s + 0.609·43-s + 2.67·45-s + 5.20·47-s − 0.291·51-s − 1.85·53-s − 3.89·55-s − 2.01·57-s − 7.29·59-s − 7.07·61-s − 1.90·65-s + ⋯ |
| L(s) = 1 | + 0.168·3-s − 0.410·5-s − 0.971·9-s + 1.27·11-s + 0.575·13-s − 0.0691·15-s − 0.242·17-s − 1.58·19-s + 0.312·23-s − 0.831·25-s − 0.331·27-s + 1.86·29-s − 0.183·31-s + 0.215·33-s − 0.994·37-s + 0.0969·39-s + 0.576·41-s + 0.0929·43-s + 0.399·45-s + 0.759·47-s − 0.0408·51-s − 0.254·53-s − 0.524·55-s − 0.267·57-s − 0.949·59-s − 0.906·61-s − 0.236·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.291T + 3T^{2} \) |
| 5 | \( 1 + 0.918T + 5T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 - 0.609T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 0.512T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 + 3.73T + 97T^{2} \) |
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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
−7.80132522362741698230367915821, −6.74367211289920604221595062397, −6.35593489256014280222999112955, −5.65311045232573893948073920483, −4.55758694560573348791636505288, −4.02290777681318132162284720402, −3.23617792401126641181271615728, −2.34464665142662653094312440546, −1.29543355651748872993832356696, 0,
1.29543355651748872993832356696, 2.34464665142662653094312440546, 3.23617792401126641181271615728, 4.02290777681318132162284720402, 4.55758694560573348791636505288, 5.65311045232573893948073920483, 6.35593489256014280222999112955, 6.74367211289920604221595062397, 7.80132522362741698230367915821
