| L(s) = 1 | + 5-s + 9-s − 2·29-s − 5·37-s + 2·41-s + 45-s − 4·49-s − 5·53-s + 2·61-s + 3·89-s − 2·101-s + 2·109-s + 5·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 5-s + 9-s − 2·29-s − 5·37-s + 2·41-s + 45-s − 4·49-s − 5·53-s + 2·61-s + 3·89-s − 2·101-s + 2·109-s + 5·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5631873657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5631873657\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.489396176711942148396774653560, −7.978463847911908916938723159038, −7.84414524235281552958243320580, −7.71483788247822969985118280640, −7.60550967556955935521421202044, −7.05623045946663616135209441725, −6.74549520858252794415983061086, −6.73897551594214695699563904732, −6.52427110502955273373945286963, −6.23331572366010476200713015836, −5.75806364247508604987487712696, −5.56969426992634278410032912524, −5.48424688781297726805139371174, −5.06067267484679542299630247484, −4.65983919067586011978452406047, −4.60856107789088923068857021480, −4.40285810040452310654867563553, −3.66106938071268329555122306825, −3.43044045478236439737860899365, −3.28986863138439458539880391728, −3.18756660000288309152276272385, −2.16399591947852327969843929509, −2.04366977008997274103467555820, −1.67747576978274899193502685885, −1.51592311804360774136872033011,
1.51592311804360774136872033011, 1.67747576978274899193502685885, 2.04366977008997274103467555820, 2.16399591947852327969843929509, 3.18756660000288309152276272385, 3.28986863138439458539880391728, 3.43044045478236439737860899365, 3.66106938071268329555122306825, 4.40285810040452310654867563553, 4.60856107789088923068857021480, 4.65983919067586011978452406047, 5.06067267484679542299630247484, 5.48424688781297726805139371174, 5.56969426992634278410032912524, 5.75806364247508604987487712696, 6.23331572366010476200713015836, 6.52427110502955273373945286963, 6.73897551594214695699563904732, 6.74549520858252794415983061086, 7.05623045946663616135209441725, 7.60550967556955935521421202044, 7.71483788247822969985118280640, 7.84414524235281552958243320580, 7.978463847911908916938723159038, 8.489396176711942148396774653560
