L(s) = 1 | + 2.19·3-s + 5-s + 7-s + 1.83·9-s + 11-s − 5.64·13-s + 2.19·15-s − 4.19·17-s − 7.52·19-s + 2.19·21-s − 7.79·23-s + 25-s − 2.55·27-s + 0.894·29-s + 5.11·31-s + 2.19·33-s + 35-s + 0.287·37-s − 12.4·39-s + 5.43·41-s − 2.68·43-s + 1.83·45-s − 7.00·47-s + 49-s − 9.23·51-s + 13.6·53-s + 55-s + ⋯ |
L(s) = 1 | + 1.27·3-s + 0.447·5-s + 0.377·7-s + 0.613·9-s + 0.301·11-s − 1.56·13-s + 0.568·15-s − 1.01·17-s − 1.72·19-s + 0.480·21-s − 1.62·23-s + 0.200·25-s − 0.491·27-s + 0.166·29-s + 0.919·31-s + 0.382·33-s + 0.169·35-s + 0.0472·37-s − 1.98·39-s + 0.849·41-s − 0.409·43-s + 0.274·45-s − 1.02·47-s + 0.142·49-s − 1.29·51-s + 1.87·53-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 | \( 1 - 2.19T + 3T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.894T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 0.287T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 2.68T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 2.69T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−7.948227347684841983622952517798, −7.07490414754164939346896055244, −6.41724631188789437087902766597, −5.59431276150047035439907213367, −4.39787953005257369345811265995, −4.27067237744869283838308886461, −2.95198074383104116210254478338, −2.28632193098149034089724657338, −1.82681950843102239594602282869, 0,
1.82681950843102239594602282869, 2.28632193098149034089724657338, 2.95198074383104116210254478338, 4.27067237744869283838308886461, 4.39787953005257369345811265995, 5.59431276150047035439907213367, 6.41724631188789437087902766597, 7.07490414754164939346896055244, 7.948227347684841983622952517798