| L(s) = 1 | − 0.476·3-s + 2.17·5-s − 2.77·9-s − 0.849·11-s − 0.662·13-s − 1.03·15-s + 5.15·17-s + 6.79·19-s − 0.144·23-s − 0.259·25-s + 2.74·27-s − 9.46·29-s − 8.50·31-s + 0.404·33-s − 7.75·37-s + 0.315·39-s + 41-s − 2.77·43-s − 6.03·45-s − 3.45·47-s − 2.45·51-s + 7.28·53-s − 1.85·55-s − 3.23·57-s − 9.84·59-s − 9.68·61-s − 1.44·65-s + ⋯ |
| L(s) = 1 | − 0.274·3-s + 0.973·5-s − 0.924·9-s − 0.256·11-s − 0.183·13-s − 0.267·15-s + 1.25·17-s + 1.55·19-s − 0.0301·23-s − 0.0518·25-s + 0.528·27-s − 1.75·29-s − 1.52·31-s + 0.0704·33-s − 1.27·37-s + 0.0504·39-s + 0.156·41-s − 0.422·43-s − 0.900·45-s − 0.504·47-s − 0.343·51-s + 1.00·53-s − 0.249·55-s − 0.428·57-s − 1.28·59-s − 1.23·61-s − 0.178·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.476T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 + 0.849T + 11T^{2} \) |
| 13 | \( 1 + 0.662T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 + 0.144T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 8.50T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 + 9.84T + 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 5.94T + 89T^{2} \) |
| 97 | \( 1 + 2.25T + 97T^{2} \) |
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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
−7.55239169940439218678550986428, −6.76199073435180474734659876305, −5.79831900170857328562912492916, −5.48858884299328645441460873703, −5.09171696992708563573225556874, −3.66783955062507637288650296975, −3.16695254044383039393636182976, −2.15504210049896158347759247795, −1.34729496384020485893342944993, 0,
1.34729496384020485893342944993, 2.15504210049896158347759247795, 3.16695254044383039393636182976, 3.66783955062507637288650296975, 5.09171696992708563573225556874, 5.48858884299328645441460873703, 5.79831900170857328562912492916, 6.76199073435180474734659876305, 7.55239169940439218678550986428
