| L(s) = 1 | − 23·3-s − 50·5-s + 98·7-s + 163·9-s + 873·11-s − 185·13-s + 1.15e3·15-s − 2.01e3·17-s − 614·19-s − 2.25e3·21-s − 1.35e3·23-s + 1.87e3·25-s − 920·27-s − 999·29-s − 9.02e3·31-s − 2.00e4·33-s − 4.90e3·35-s − 9.03e3·37-s + 4.25e3·39-s − 1.83e4·41-s − 3.97e3·43-s − 8.15e3·45-s − 2.17e4·47-s + 7.20e3·49-s + 4.64e4·51-s + 2.15e3·53-s − 4.36e4·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.894·5-s + 0.755·7-s + 0.670·9-s + 2.17·11-s − 0.303·13-s + 1.31·15-s − 1.69·17-s − 0.390·19-s − 1.11·21-s − 0.532·23-s + 3/5·25-s − 0.242·27-s − 0.220·29-s − 1.68·31-s − 3.20·33-s − 0.676·35-s − 1.08·37-s + 0.447·39-s − 1.70·41-s − 0.327·43-s − 0.599·45-s − 1.43·47-s + 3/7·49-s + 2.49·51-s + 0.105·53-s − 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 23 T + 122 p T^{2} + 23 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 873 T + 492202 T^{2} - 873 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 185 T + 639900 T^{2} + 185 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2019 T + 2857624 T^{2} + 2019 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 614 T + 5037366 T^{2} + 614 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1350 T + 8524462 T^{2} + 1350 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 999 T - 409396 p T^{2} + 999 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9020 T + 77562078 T^{2} + 9020 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9032 T + 119525334 T^{2} + 9032 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18330 T + 312431386 T^{2} + 18330 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3974 T + 254730414 T^{2} + 3974 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21759 T + 363084742 T^{2} + 21759 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2154 T - 359043374 T^{2} - 2154 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 28704 T + 1626529558 T^{2} + 28704 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12394 T + 756573162 T^{2} - 12394 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 101888 T + 5051298774 T^{2} + 101888 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 91632 T + 5074655182 T^{2} - 91632 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 35996 T + 3240285846 T^{2} + 35996 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10001 T + 5950982646 T^{2} + 10001 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 78108 T + 9353568646 T^{2} - 78108 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 33438 T + 9987409978 T^{2} - 33438 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 85751 T + 18522104208 T^{2} + 85751 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.83463569161442498465360634056, −11.52874758800278248887157469183, −11.05713449119903827145491223348, −10.89095154225061174318462550126, −10.05336794045128401626943290293, −9.322847436962971726568125571992, −8.730447031714568803298936674122, −8.495250220736395012701449769291, −7.45611805095338508134763291306, −7.07375352158757978820724226381, −6.33172512530810616449391441881, −6.18140121496176081656799128096, −5.05876045819936602431284814996, −4.80221907474165780346660874016, −3.96328467377064558699856781036, −3.56142135614041130835043343632, −1.96859187019795435627944838389, −1.39101879758937639388449068724, 0, 0,
1.39101879758937639388449068724, 1.96859187019795435627944838389, 3.56142135614041130835043343632, 3.96328467377064558699856781036, 4.80221907474165780346660874016, 5.05876045819936602431284814996, 6.18140121496176081656799128096, 6.33172512530810616449391441881, 7.07375352158757978820724226381, 7.45611805095338508134763291306, 8.495250220736395012701449769291, 8.730447031714568803298936674122, 9.322847436962971726568125571992, 10.05336794045128401626943290293, 10.89095154225061174318462550126, 11.05713449119903827145491223348, 11.52874758800278248887157469183, 11.83463569161442498465360634056
