| L(s) = 1 | − 2.64·3-s + 3.09·5-s − 1.88·7-s + 3.97·9-s + 0.597·11-s + 2.98·13-s − 8.18·15-s − 17-s + 4.89·19-s + 4.96·21-s − 4.83·23-s + 4.60·25-s − 2.56·27-s − 2.75·29-s − 5.24·31-s − 1.57·33-s − 5.83·35-s + 1.62·37-s − 7.87·39-s + 0.567·41-s − 4.98·43-s + 12.3·45-s − 13.6·47-s − 3.45·49-s + 2.64·51-s + 12.5·53-s + 1.85·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 1.38·5-s − 0.711·7-s + 1.32·9-s + 0.180·11-s + 0.827·13-s − 2.11·15-s − 0.242·17-s + 1.12·19-s + 1.08·21-s − 1.00·23-s + 0.920·25-s − 0.492·27-s − 0.511·29-s − 0.942·31-s − 0.274·33-s − 0.985·35-s + 0.267·37-s − 1.26·39-s + 0.0885·41-s − 0.759·43-s + 1.83·45-s − 1.98·47-s − 0.494·49-s + 0.369·51-s + 1.72·53-s + 0.249·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 0.597T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 0.567T + 41T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 0.921T + 71T^{2} \) |
| 73 | \( 1 + 6.23T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.12375757439285199592346562118, −6.51602166871898276042018666278, −6.05719680739430653204659828916, −5.55849140647726133597710262031, −5.03609758234616574144863625298, −3.99712062682214909267280276917, −3.13970582834571286313667619086, −1.94557337635968248488608538771, −1.20490125433407660414199199464, 0,
1.20490125433407660414199199464, 1.94557337635968248488608538771, 3.13970582834571286313667619086, 3.99712062682214909267280276917, 5.03609758234616574144863625298, 5.55849140647726133597710262031, 6.05719680739430653204659828916, 6.51602166871898276042018666278, 7.12375757439285199592346562118
