| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 3·9-s + 2·11-s + 2·13-s + 4·15-s − 2·17-s + 4·21-s − 8·23-s + 3·25-s − 4·27-s + 4·29-s − 4·33-s + 4·35-s + 12·37-s − 4·39-s + 12·41-s + 2·43-s − 6·45-s + 8·47-s + 6·49-s + 4·51-s − 4·55-s + 8·59-s + 20·61-s − 6·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.872·21-s − 1.66·23-s + 3/5·25-s − 0.769·27-s + 0.742·29-s − 0.696·33-s + 0.676·35-s + 1.97·37-s − 0.640·39-s + 1.87·41-s + 0.304·43-s − 0.894·45-s + 1.16·47-s + 6/7·49-s + 0.560·51-s − 0.539·55-s + 1.04·59-s + 2.56·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118216115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118216115\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.772311302568421604563812528836, −9.660453784589624343922139692012, −9.063147881719105273684475345596, −8.554052647885741308328579466048, −8.213274300683371859981225652762, −7.81340805931322677559658006624, −7.12290540691687084793027189216, −7.09668252075945322943668616017, −6.31302706333399249345648539357, −6.29219006546869943706312695738, −5.67288531197309155059925688884, −5.45471202853487126074755065994, −4.41954484398844085053880964208, −4.41501320991189656430309456799, −3.86572890316638213149104375862, −3.55801716397703883927008569935, −2.58041316894354798002629074743, −2.20441821729702906718946433058, −0.976174657239473000499428641341, −0.62247197613638321999312028871,
0.62247197613638321999312028871, 0.976174657239473000499428641341, 2.20441821729702906718946433058, 2.58041316894354798002629074743, 3.55801716397703883927008569935, 3.86572890316638213149104375862, 4.41501320991189656430309456799, 4.41954484398844085053880964208, 5.45471202853487126074755065994, 5.67288531197309155059925688884, 6.29219006546869943706312695738, 6.31302706333399249345648539357, 7.09668252075945322943668616017, 7.12290540691687084793027189216, 7.81340805931322677559658006624, 8.213274300683371859981225652762, 8.554052647885741308328579466048, 9.063147881719105273684475345596, 9.660453784589624343922139692012, 9.772311302568421604563812528836
