L(s) = 1 | + 5·2-s + 9·3-s + 16·4-s − 3·5-s + 45·6-s − 91·7-s + 115·8-s + 54·9-s − 15·10-s + 149·11-s + 144·12-s − 455·14-s − 27·15-s + 575·16-s − 462·17-s + 270·18-s − 618·19-s − 48·20-s − 819·21-s + 745·22-s + 560·23-s + 1.03e3·24-s − 619·25-s + 243·27-s − 1.45e3·28-s + 470·29-s − 135·30-s + ⋯ |
L(s) = 1 | + 5/4·2-s + 3-s + 4-s − 0.119·5-s + 5/4·6-s − 1.85·7-s + 1.79·8-s + 2/3·9-s − 0.149·10-s + 1.23·11-s + 12-s − 2.32·14-s − 0.119·15-s + 2.24·16-s − 1.59·17-s + 5/6·18-s − 1.71·19-s − 0.119·20-s − 1.85·21-s + 1.53·22-s + 1.05·23-s + 1.79·24-s − 0.990·25-s + 1/3·27-s − 1.85·28-s + 0.558·29-s − 0.149·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.452132357\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.452132357\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | $C_2$ | \( 1 + 13 p T + p^{4} T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 5 T + 9 T^{2} - 5 p^{4} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 628 T^{2} + 3 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 149 T + 7560 T^{2} - 149 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 55394 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 462 T + 154669 T^{2} + 462 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 618 T + 257629 T^{2} + 618 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 560 T + 33759 T^{2} - 560 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 235 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2229 T + 2579668 T^{2} + 2229 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 1970 T + 2006739 T^{2} + 1970 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2280106 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2798 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7152 T + 21930049 T^{2} - 7152 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 17 p T - 2520 p^{2} T^{2} + 17 p^{5} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2391 T + 14022988 T^{2} - 2391 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6180 T + 26576641 T^{2} + 6180 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4156 T - 2878785 T^{2} - 4156 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 484 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1644 T + 29299153 T^{2} + 1644 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 4325 T - 20244456 T^{2} + 4325 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 93215615 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3426 T + 66654733 T^{2} - 3426 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 147347735 T^{2} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.26071596033061159917811024032, −17.06353876396204951004320823079, −16.14672090410641187392522148220, −15.61037131754701026876493797503, −15.17891062931089409339460142661, −14.19048938470903591684688158815, −13.96373222339278553526734980912, −13.08901447454589575230017439325, −12.84236913370184628840375047930, −12.23533153727057564348312320574, −10.85902140365596966638594164479, −10.53789861929356622909581032783, −9.111807104244553977633501411992, −9.017840311828740369852225580106, −7.35567519262057096249294130785, −6.85229238609290911030047716372, −5.91502003178006900602282358491, −4.15887196620337924832531694878, −3.83444810210489406457311791242, −2.32499222925034976875112394581,
2.32499222925034976875112394581, 3.83444810210489406457311791242, 4.15887196620337924832531694878, 5.91502003178006900602282358491, 6.85229238609290911030047716372, 7.35567519262057096249294130785, 9.017840311828740369852225580106, 9.111807104244553977633501411992, 10.53789861929356622909581032783, 10.85902140365596966638594164479, 12.23533153727057564348312320574, 12.84236913370184628840375047930, 13.08901447454589575230017439325, 13.96373222339278553526734980912, 14.19048938470903591684688158815, 15.17891062931089409339460142661, 15.61037131754701026876493797503, 16.14672090410641187392522148220, 17.06353876396204951004320823079, 17.26071596033061159917811024032