L(s) = 1 | + 0.445·3-s − 2.55·5-s + 3.85·7-s − 2.80·9-s − 5.54·11-s + 0.692·13-s − 1.13·15-s + 5.04·17-s + 1.08·19-s + 1.71·21-s + 7.80·23-s + 1.52·25-s − 2.58·27-s − 10.1·29-s + 7.65·31-s − 2.46·33-s − 9.83·35-s + 3.51·37-s + 0.307·39-s − 9.38·41-s − 5.85·43-s + 7.15·45-s − 0.841·47-s + 7.82·49-s + 2.24·51-s − 2.77·53-s + 14.1·55-s + ⋯ |
L(s) = 1 | + 0.256·3-s − 1.14·5-s + 1.45·7-s − 0.933·9-s − 1.67·11-s + 0.191·13-s − 0.293·15-s + 1.22·17-s + 0.249·19-s + 0.373·21-s + 1.62·23-s + 0.305·25-s − 0.496·27-s − 1.88·29-s + 1.37·31-s − 0.429·33-s − 1.66·35-s + 0.577·37-s + 0.0493·39-s − 1.46·41-s − 0.892·43-s + 1.06·45-s − 0.122·47-s + 1.11·49-s + 0.314·51-s − 0.381·53-s + 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 5.54T + 11T^{2} \) |
| 13 | \( 1 - 0.692T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 0.841T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 - 3.52T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 0.567T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.118310711270485393083919115068, −7.78059111287612426292968828205, −7.10889875961380777721909619952, −5.63290228107926513794920852870, −5.22451708597556623165449418687, −4.45910297750454158790064800756, −3.33786000820537713492813595678, −2.75801306075941120984241266024, −1.43353711104688231527171445554, 0,
1.43353711104688231527171445554, 2.75801306075941120984241266024, 3.33786000820537713492813595678, 4.45910297750454158790064800756, 5.22451708597556623165449418687, 5.63290228107926513794920852870, 7.10889875961380777721909619952, 7.78059111287612426292968828205, 8.118310711270485393083919115068