L(s) = 1 | + 2.23·2-s − 3-s + 3.00·4-s − 0.618·5-s − 2.23·6-s + 7-s + 2.23·8-s + 9-s − 1.38·10-s − 3.00·12-s + 3.23·13-s + 2.23·14-s + 0.618·15-s − 0.999·16-s + 4.85·17-s + 2.23·18-s + 2.85·19-s − 1.85·20-s − 21-s + 4.38·23-s − 2.23·24-s − 4.61·25-s + 7.23·26-s − 27-s + 3.00·28-s − 6·29-s + 1.38·30-s + ⋯ |
L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.276·5-s − 0.912·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.437·10-s − 0.866·12-s + 0.897·13-s + 0.597·14-s + 0.159·15-s − 0.249·16-s + 1.17·17-s + 0.527·18-s + 0.654·19-s − 0.414·20-s − 0.218·21-s + 0.913·23-s − 0.456·24-s − 0.923·25-s + 1.41·26-s − 0.192·27-s + 0.566·28-s − 1.11·29-s + 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.952824589\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.952824589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.935277039265100822582943894821, −7.70203729227585032319553505324, −7.28552522292009437632091066648, −6.14791439071141835084673715635, −5.66853214167679404956862056357, −5.07266603886371937056629384492, −4.03609835182717613984766672393, −3.60579504317032388492675763627, −2.45242438535002073366048705990, −1.09521240941419714104075119347,
1.09521240941419714104075119347, 2.45242438535002073366048705990, 3.60579504317032388492675763627, 4.03609835182717613984766672393, 5.07266603886371937056629384492, 5.66853214167679404956862056357, 6.14791439071141835084673715635, 7.28552522292009437632091066648, 7.70203729227585032319553505324, 8.935277039265100822582943894821