| L(s) = 1 | + 2.74·3-s − 0.391·5-s − 2.92·7-s + 4.51·9-s − 6.05·11-s + 5.42·13-s − 1.07·15-s − 5.02·17-s + 19-s − 8.02·21-s + 7.86·23-s − 4.84·25-s + 4.14·27-s − 4.98·29-s − 5.55·31-s − 16.5·33-s + 1.14·35-s − 6.33·37-s + 14.8·39-s − 8.82·41-s + 1.17·43-s − 1.76·45-s + 0.240·47-s + 1.57·49-s − 13.7·51-s − 53-s + 2.36·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s − 0.174·5-s − 1.10·7-s + 1.50·9-s − 1.82·11-s + 1.50·13-s − 0.276·15-s − 1.21·17-s + 0.229·19-s − 1.75·21-s + 1.64·23-s − 0.969·25-s + 0.796·27-s − 0.926·29-s − 0.997·31-s − 2.88·33-s + 0.193·35-s − 1.04·37-s + 2.38·39-s − 1.37·41-s + 0.178·43-s − 0.263·45-s + 0.0350·47-s + 0.224·49-s − 1.92·51-s − 0.137·53-s + 0.319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 + 0.391T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + 6.33T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 - 0.240T + 47T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 - 2.22T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 3.64T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 2.94T + 97T^{2} \) |
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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
−8.274720076731247767321571551241, −7.40532670816638107390121025158, −6.90645015630698568159353271790, −5.88255173850057468772752747762, −5.01873511646788384203127851776, −3.82510585256694332145116372737, −3.35418892813026985097475369812, −2.66815449189576057660845758430, −1.76372522666884682337439728321, 0,
1.76372522666884682337439728321, 2.66815449189576057660845758430, 3.35418892813026985097475369812, 3.82510585256694332145116372737, 5.01873511646788384203127851776, 5.88255173850057468772752747762, 6.90645015630698568159353271790, 7.40532670816638107390121025158, 8.274720076731247767321571551241
