L(s) = 1 | + (0.235 − 0.971i)2-s + (0.252 + 1.75i)3-s + (−0.888 − 0.458i)4-s + (0.415 − 0.909i)5-s + (1.76 + 0.168i)6-s + (1.34 + 1.28i)7-s + (−0.654 + 0.755i)8-s + (−2.07 + 0.608i)9-s + (−0.786 − 0.618i)10-s + (0.581 − 1.67i)12-s + (1.56 − 1.00i)14-s + (1.70 + 0.500i)15-s + (0.580 + 0.814i)16-s + (0.102 + 2.15i)18-s + (−0.786 + 0.618i)20-s + (−1.91 + 2.68i)21-s + ⋯ |
L(s) = 1 | + (0.235 − 0.971i)2-s + (0.252 + 1.75i)3-s + (−0.888 − 0.458i)4-s + (0.415 − 0.909i)5-s + (1.76 + 0.168i)6-s + (1.34 + 1.28i)7-s + (−0.654 + 0.755i)8-s + (−2.07 + 0.608i)9-s + (−0.786 − 0.618i)10-s + (0.581 − 1.67i)12-s + (1.56 − 1.00i)14-s + (1.70 + 0.500i)15-s + (0.580 + 0.814i)16-s + (0.102 + 2.15i)18-s + (−0.786 + 0.618i)20-s + (−1.91 + 2.68i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.368308201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368308201\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.235 + 0.971i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.0475 - 0.998i)T \) |
good | 3 | \( 1 + (-0.252 - 1.75i)T + (-0.959 + 0.281i)T^{2} \) |
| 7 | \( 1 + (-1.34 - 1.28i)T + (0.0475 + 0.998i)T^{2} \) |
| 11 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 13 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 17 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 19 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 23 | \( 1 + (0.928 + 0.371i)T + (0.723 + 0.690i)T^{2} \) |
| 29 | \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.0934 + 1.96i)T + (-0.995 - 0.0950i)T^{2} \) |
| 43 | \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.514 + 0.404i)T + (0.235 - 0.971i)T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.827 + 0.0789i)T + (0.981 + 0.189i)T^{2} \) |
| 71 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 73 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 79 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 83 | \( 1 + (-0.839 - 1.17i)T + (-0.327 + 0.945i)T^{2} \) |
| 89 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.938452803168832548023412984277, −8.978320633841033191012097818186, −8.815967477388806935035023077207, −8.137497366046613427754257289244, −5.78714367130296226772044865402, −5.32370401967610376577225058026, −4.69131924694994237710092815603, −4.02351527897609514428383924378, −2.76798226930760143304906187787, −1.83065906616659269374068496262,
1.20922598231804179553607016683, 2.43425111801727657839615423428, 3.72943825850722333907750361549, 4.87366314240502644974893482311, 6.12116618226899675048541463740, 6.50341742959205792804803288576, 7.53602969862797236481437602350, 7.69501306251626218306063958351, 8.346442702890765528002610979590, 9.583438866107151967723490069430