L(s) = 1 | + 0.406·3-s − 5-s + 7-s − 2.83·9-s − 11-s − 0.813·13-s − 0.406·15-s + 3.83·17-s − 5.42·19-s + 0.406·21-s + 5.42·23-s + 25-s − 2.37·27-s + 6.61·29-s − 6·31-s − 0.406·33-s − 35-s + 8.24·37-s − 0.330·39-s − 5.59·41-s + 3.02·43-s + 2.83·45-s + 5.02·47-s + 49-s + 1.55·51-s + 6.61·53-s + 55-s + ⋯ |
L(s) = 1 | + 0.234·3-s − 0.447·5-s + 0.377·7-s − 0.944·9-s − 0.301·11-s − 0.225·13-s − 0.105·15-s + 0.930·17-s − 1.24·19-s + 0.0887·21-s + 1.13·23-s + 0.200·25-s − 0.456·27-s + 1.22·29-s − 1.07·31-s − 0.0708·33-s − 0.169·35-s + 1.35·37-s − 0.0529·39-s − 0.873·41-s + 0.460·43-s + 0.422·45-s + 0.732·47-s + 0.142·49-s + 0.218·51-s + 0.908·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.406T + 3T^{2} \) |
| 13 | \( 1 + 0.813T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.978T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.20T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−7.70387526781146541595835537070, −7.20335073256615762864299720899, −6.21065334130404678339569845975, −5.57558186585278716580770516167, −4.78047824841178686831854018554, −4.06643059688311044025715231231, −3.05830390514459087216783851728, −2.54166462851648834844585836093, −1.28439028395718501629323165446, 0,
1.28439028395718501629323165446, 2.54166462851648834844585836093, 3.05830390514459087216783851728, 4.06643059688311044025715231231, 4.78047824841178686831854018554, 5.57558186585278716580770516167, 6.21065334130404678339569845975, 7.20335073256615762864299720899, 7.70387526781146541595835537070