L(s) = 1 | − 3.23·3-s + 2·5-s + 7-s + 7.47·9-s + 11-s + 1.23·13-s − 6.47·15-s + 1.23·17-s + 2.47·19-s − 3.23·21-s − 6.47·23-s − 25-s − 14.4·27-s + 0.472·29-s − 7.23·31-s − 3.23·33-s + 2·35-s − 0.472·37-s − 4.00·39-s − 6.76·41-s − 8·43-s + 14.9·45-s + 7.23·47-s + 49-s − 4.00·51-s − 8.47·53-s + 2·55-s + ⋯ |
L(s) = 1 | − 1.86·3-s + 0.894·5-s + 0.377·7-s + 2.49·9-s + 0.301·11-s + 0.342·13-s − 1.67·15-s + 0.299·17-s + 0.567·19-s − 0.706·21-s − 1.34·23-s − 0.200·25-s − 2.78·27-s + 0.0876·29-s − 1.29·31-s − 0.563·33-s + 0.338·35-s − 0.0776·37-s − 0.640·39-s − 1.05·41-s − 1.21·43-s + 2.22·45-s + 1.05·47-s + 0.142·49-s − 0.560·51-s − 1.16·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−7.64729643138933454798838607709, −6.97405614955225443292224702570, −6.18474237996884080521406510545, −5.76641894505025388411227895443, −5.20581496474394418506265721901, −4.42882364422748721976030400842, −3.54280677722574907035663580031, −1.92187814535725595304736950897, −1.33054705729530024331368261493, 0,
1.33054705729530024331368261493, 1.92187814535725595304736950897, 3.54280677722574907035663580031, 4.42882364422748721976030400842, 5.20581496474394418506265721901, 5.76641894505025388411227895443, 6.18474237996884080521406510545, 6.97405614955225443292224702570, 7.64729643138933454798838607709