| L(s) = 1 | + 3.23·3-s − 1.23·7-s + 7.47·9-s − 2·11-s − 4.47·13-s + 4.47·17-s + 4.47·19-s − 4.00·21-s + 9.23·23-s + 14.4·27-s − 2·29-s − 2.47·31-s − 6.47·33-s + 10.9·37-s − 14.4·39-s + 3.52·41-s − 5.70·43-s + 2.76·47-s − 5.47·49-s + 14.4·51-s + 8.47·53-s + 14.4·57-s + 0.472·59-s − 6·61-s − 9.23·63-s + 5.70·67-s + 29.8·69-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 0.467·7-s + 2.49·9-s − 0.603·11-s − 1.24·13-s + 1.08·17-s + 1.02·19-s − 0.872·21-s + 1.92·23-s + 2.78·27-s − 0.371·29-s − 0.444·31-s − 1.12·33-s + 1.79·37-s − 2.31·39-s + 0.550·41-s − 0.870·43-s + 0.403·47-s − 0.781·49-s + 2.02·51-s + 1.16·53-s + 1.91·57-s + 0.0614·59-s − 0.768·61-s − 1.16·63-s + 0.697·67-s + 3.59·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.713040626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.713040626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−8.702359391795760127122012866109, −7.77742646207257115461543100573, −7.50579430311463550537508283164, −6.80624259319343797894104553597, −5.44568036984658284447605357533, −4.70180550461449963703163615525, −3.62421041640382262405841866570, −2.95528857473292294870008251077, −2.44203837666168113250173195072, −1.12891604678039465084558040210,
1.12891604678039465084558040210, 2.44203837666168113250173195072, 2.95528857473292294870008251077, 3.62421041640382262405841866570, 4.70180550461449963703163615525, 5.44568036984658284447605357533, 6.80624259319343797894104553597, 7.50579430311463550537508283164, 7.77742646207257115461543100573, 8.702359391795760127122012866109
