| L(s) = 1 | − 5·2-s + 7·4-s − 15·5-s − 15·7-s + 15·8-s + 75·10-s + 17·11-s + 75·14-s − 65·16-s − 70·17-s + 141·19-s − 105·20-s − 85·22-s − 145·23-s + 25·25-s − 105·28-s − 34·29-s + 140·31-s + 95·32-s + 350·34-s + 225·35-s + 190·37-s − 705·38-s − 225·40-s + 538·41-s + 455·43-s + 119·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 7/8·4-s − 1.34·5-s − 0.809·7-s + 0.662·8-s + 2.37·10-s + 0.465·11-s + 1.43·14-s − 1.01·16-s − 0.998·17-s + 1.70·19-s − 1.17·20-s − 0.823·22-s − 1.31·23-s + 1/5·25-s − 0.708·28-s − 0.217·29-s + 0.811·31-s + 0.524·32-s + 1.76·34-s + 1.08·35-s + 0.844·37-s − 3.00·38-s − 0.889·40-s + 2.04·41-s + 1.61·43-s + 0.407·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 5 T + 9 p T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 p T + 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 15 T + 738 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 17 T + 1778 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 70 T + 9963 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 141 T + 978 p T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 145 T + 24962 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 34 T + 33767 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 140 T + 21982 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 190 T + 102103 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 538 T + 208503 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 455 T + 170782 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 60 T + 125246 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 545 T + 256304 T^{2} + 545 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 809 T + 561522 T^{2} + 809 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 502 T + 346963 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 475 T + 622738 T^{2} - 475 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 127 T + 672998 T^{2} - 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 585 T + 832884 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 260 T + 1117974 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 921 T + 1555592 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 415 T + 933568 T^{2} - 415 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.028846860301285114815939076465, −8.492553806430241363141624930796, −8.040624311993847007093685666135, −7.895243051377449855482698446461, −7.40662131209223513604549297313, −7.38098639561304497483699160589, −6.45576384966031824968973947886, −6.31181112906464042209212185517, −5.82723221069999670170110774503, −5.13101321923083465182337117206, −4.37589210240560850595647254091, −4.36407617482326653610571373176, −3.74906813297077226033666012331, −3.25886754824020089737239112160, −2.70143500491642066022922190434, −2.03329052862266446654586087612, −1.14480537131054448136326617302, −0.848997921979859223264229607217, 0, 0,
0.848997921979859223264229607217, 1.14480537131054448136326617302, 2.03329052862266446654586087612, 2.70143500491642066022922190434, 3.25886754824020089737239112160, 3.74906813297077226033666012331, 4.36407617482326653610571373176, 4.37589210240560850595647254091, 5.13101321923083465182337117206, 5.82723221069999670170110774503, 6.31181112906464042209212185517, 6.45576384966031824968973947886, 7.38098639561304497483699160589, 7.40662131209223513604549297313, 7.895243051377449855482698446461, 8.040624311993847007093685666135, 8.492553806430241363141624930796, 9.028846860301285114815939076465
