| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 2·9-s − 6·11-s + 4·12-s + 6·13-s − 4·16-s + 8·17-s + 4·18-s + 2·19-s − 12·22-s + 12·26-s + 6·27-s − 6·29-s − 8·32-s − 12·33-s + 16·34-s + 4·36-s + 6·37-s + 4·38-s + 12·39-s − 6·43-s − 12·44-s + 4·47-s − 8·48-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 2/3·9-s − 1.80·11-s + 1.15·12-s + 1.66·13-s − 16-s + 1.94·17-s + 0.942·18-s + 0.458·19-s − 2.55·22-s + 2.35·26-s + 1.15·27-s − 1.11·29-s − 1.41·32-s − 2.08·33-s + 2.74·34-s + 2/3·36-s + 0.986·37-s + 0.648·38-s + 1.92·39-s − 0.914·43-s − 1.80·44-s + 0.583·47-s − 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.060207881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.060207881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−11.83235190011111378327495194270, −10.97331924486673481772876763146, −10.56603066080485562666516625863, −10.41937870001261896317005151130, −9.579682634459839304782542282060, −9.225141314365186510454936096481, −8.678367616028273321787463831652, −8.166388762424466795394293747641, −7.83622107445758174487142653723, −7.35760235225021936419799915991, −6.80159913866874442669604157919, −5.83478571311935492161291049131, −5.77794826452728323459459877510, −5.25436762822514581017161902915, −4.57138035523111835632807934280, −3.78185485567728945401117648064, −3.55091391766949795227568733252, −2.79855823225857885680085386947, −2.55412636520891000078938723688, −1.29289217716584487727594934369,
1.29289217716584487727594934369, 2.55412636520891000078938723688, 2.79855823225857885680085386947, 3.55091391766949795227568733252, 3.78185485567728945401117648064, 4.57138035523111835632807934280, 5.25436762822514581017161902915, 5.77794826452728323459459877510, 5.83478571311935492161291049131, 6.80159913866874442669604157919, 7.35760235225021936419799915991, 7.83622107445758174487142653723, 8.166388762424466795394293747641, 8.678367616028273321787463831652, 9.225141314365186510454936096481, 9.579682634459839304782542282060, 10.41937870001261896317005151130, 10.56603066080485562666516625863, 10.97331924486673481772876763146, 11.83235190011111378327495194270
