| L(s) = 1 | − 3.37·3-s + 3.80·5-s + 2.38·7-s + 8.41·9-s + 3.47·11-s + 1.53·13-s − 12.8·15-s − 2.01·19-s − 8.07·21-s − 5.51·23-s + 9.46·25-s − 18.3·27-s − 1.09·29-s − 0.127·31-s − 11.7·33-s + 9.08·35-s + 7.21·37-s − 5.17·39-s − 6.48·41-s + 6.14·43-s + 32.0·45-s − 2.62·47-s − 1.29·49-s − 4.32·53-s + 13.2·55-s + 6.80·57-s − 12.6·59-s + ⋯ |
| L(s) = 1 | − 1.95·3-s + 1.70·5-s + 0.902·7-s + 2.80·9-s + 1.04·11-s + 0.424·13-s − 3.31·15-s − 0.462·19-s − 1.76·21-s − 1.14·23-s + 1.89·25-s − 3.52·27-s − 0.202·29-s − 0.0228·31-s − 2.04·33-s + 1.53·35-s + 1.18·37-s − 0.828·39-s − 1.01·41-s + 0.937·43-s + 4.77·45-s − 0.383·47-s − 0.185·49-s − 0.594·53-s + 1.78·55-s + 0.901·57-s − 1.64·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979561038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979561038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 0.127T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 6.14T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 + 4.32T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 0.710T + 67T^{2} \) |
| 71 | \( 1 - 7.38T + 71T^{2} \) |
| 73 | \( 1 - 2.41T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 5.02T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.47603532639777650185311322721, −6.58176825227992147018447466054, −6.21476581640086903806597050566, −5.84506955544558258252065198577, −5.05533915773480123866319171052, −4.59736337536562123117637144773, −3.75293924929365154158941665648, −2.06944129618394607474322355548, −1.61729658528538040322900515857, −0.812482392620028408769671201232,
0.812482392620028408769671201232, 1.61729658528538040322900515857, 2.06944129618394607474322355548, 3.75293924929365154158941665648, 4.59736337536562123117637144773, 5.05533915773480123866319171052, 5.84506955544558258252065198577, 6.21476581640086903806597050566, 6.58176825227992147018447466054, 7.47603532639777650185311322721
