| L(s) = 1 | − 2·25-s − 48·47-s − 28·49-s + 8·73-s + 40·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 2/5·25-s − 7.00·47-s − 4·49-s + 0.936·73-s + 4.06·97-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.247746039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247746039\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−6.20951248550338930300086343201, −6.11439537048361207806173410786, −5.82924586667774568763155563454, −5.76696242484838367152798992846, −5.18683837436781675448449307285, −5.17246276347372053012988691613, −4.91388260307736756185377043723, −4.77505670274007989256175620193, −4.70847182709817942351490025705, −4.38708068752849014183241000406, −4.37861499620340767171492589987, −3.64831149060537691790954595544, −3.51507474476073292934832537157, −3.46514764749875631524545818920, −3.33658233123524629953233622899, −3.17126645367888624184143647547, −2.81303029260769667657205135983, −2.40646412631811534674645934426, −2.19772310714548487395609426214, −1.86244454261895779495460442998, −1.68591569063055145264997413258, −1.43634286356530895594028325308, −1.22735761786988618322948324704, −0.49699874778470594520394715810, −0.23417480768093765063143128891,
0.23417480768093765063143128891, 0.49699874778470594520394715810, 1.22735761786988618322948324704, 1.43634286356530895594028325308, 1.68591569063055145264997413258, 1.86244454261895779495460442998, 2.19772310714548487395609426214, 2.40646412631811534674645934426, 2.81303029260769667657205135983, 3.17126645367888624184143647547, 3.33658233123524629953233622899, 3.46514764749875631524545818920, 3.51507474476073292934832537157, 3.64831149060537691790954595544, 4.37861499620340767171492589987, 4.38708068752849014183241000406, 4.70847182709817942351490025705, 4.77505670274007989256175620193, 4.91388260307736756185377043723, 5.17246276347372053012988691613, 5.18683837436781675448449307285, 5.76696242484838367152798992846, 5.82924586667774568763155563454, 6.11439537048361207806173410786, 6.20951248550338930300086343201
