| L(s) = 1 | − 0.456·2-s − 2.79·3-s − 1.79·4-s + 0.456·5-s + 1.27·6-s + 1.73·8-s + 4.79·9-s − 0.208·10-s + 3.92·11-s + 5·12-s − 1.27·15-s + 2.79·16-s + 3·17-s − 2.18·18-s + 1.37·19-s − 0.818·20-s − 1.79·22-s + 1.58·23-s − 4.83·24-s − 4.79·25-s − 4.99·27-s − 6.79·29-s + 0.582·30-s + 8.66·31-s − 4.73·32-s − 10.9·33-s − 1.37·34-s + ⋯ |
| L(s) = 1 | − 0.323·2-s − 1.61·3-s − 0.895·4-s + 0.204·5-s + 0.520·6-s + 0.612·8-s + 1.59·9-s − 0.0660·10-s + 1.18·11-s + 1.44·12-s − 0.329·15-s + 0.697·16-s + 0.727·17-s − 0.515·18-s + 0.314·19-s − 0.182·20-s − 0.381·22-s + 0.329·23-s − 0.986·24-s − 0.958·25-s − 0.962·27-s − 1.26·29-s + 0.106·30-s + 1.55·31-s − 0.837·32-s − 1.90·33-s − 0.235·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.456T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 0.456T + 5T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 - 8.66T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 7.84T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 9.57T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−7.24809860003143351199647333564, −6.78149982352485609127046640131, −5.85610787012836845684321359523, −5.51948046883494235823935566341, −4.77173095065951897202247392764, −4.09987537926902573749292591649, −3.36950479475311550506925779534, −1.68368053573914757319938666775, −1.01324135603874719641719468789, 0,
1.01324135603874719641719468789, 1.68368053573914757319938666775, 3.36950479475311550506925779534, 4.09987537926902573749292591649, 4.77173095065951897202247392764, 5.51948046883494235823935566341, 5.85610787012836845684321359523, 6.78149982352485609127046640131, 7.24809860003143351199647333564
