| L(s) = 1 | + 4·3-s + 2·5-s + 4·7-s + 8·9-s + 4·11-s − 4·13-s + 8·15-s − 4·17-s + 2·19-s + 16·21-s + 12·23-s + 3·25-s + 12·27-s + 4·29-s + 8·31-s + 16·33-s + 8·35-s − 12·37-s − 16·39-s − 4·41-s − 4·43-s + 16·45-s + 4·47-s + 6·49-s − 16·51-s + 4·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s + 1.20·11-s − 1.10·13-s + 2.06·15-s − 0.970·17-s + 0.458·19-s + 3.49·21-s + 2.50·23-s + 3/5·25-s + 2.30·27-s + 0.742·29-s + 1.43·31-s + 2.78·33-s + 1.35·35-s − 1.97·37-s − 2.56·39-s − 0.624·41-s − 0.609·43-s + 2.38·45-s + 0.583·47-s + 6/7·49-s − 2.24·51-s + 0.549·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.325469940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.325469940\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.373346786878501293642312781848, −9.126219677021526773619159829210, −8.889932070403171249287295323269, −8.631701213045925488478188799566, −8.181895377243302681506538839004, −7.83667600992098535447578319865, −7.30155095873499276115561178340, −7.04007419226724166778727163224, −6.55523169887341825854139924299, −6.23951813168676526941690728695, −5.13237721787974012480990535454, −5.07443515478179354900070296507, −4.64366623492648899300369469242, −4.22125217468796483542276910683, −3.35359836181160138157132269303, −3.14706206559333232854953325503, −2.55556377588078431300180226095, −2.22346747470726762631896537131, −1.48431367344935190912877200754, −1.26019537507612455551548251757,
1.26019537507612455551548251757, 1.48431367344935190912877200754, 2.22346747470726762631896537131, 2.55556377588078431300180226095, 3.14706206559333232854953325503, 3.35359836181160138157132269303, 4.22125217468796483542276910683, 4.64366623492648899300369469242, 5.07443515478179354900070296507, 5.13237721787974012480990535454, 6.23951813168676526941690728695, 6.55523169887341825854139924299, 7.04007419226724166778727163224, 7.30155095873499276115561178340, 7.83667600992098535447578319865, 8.181895377243302681506538839004, 8.631701213045925488478188799566, 8.889932070403171249287295323269, 9.126219677021526773619159829210, 9.373346786878501293642312781848
