| L(s) = 1 | − 24·23-s + 4·25-s + 24·47-s + 26·49-s + 48·71-s + 28·73-s + 68·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.00·23-s + 4/5·25-s + 3.50·47-s + 26/7·49-s + 5.69·71-s + 3.27·73-s + 6.90·97-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 + 3.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.842449995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.842449995\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{4} \) |
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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
−6.56809349723715479604337229923, −6.34421449466675778362521060961, −6.21905585924481829084851771926, −6.00008704750756131241545925467, −5.87929871135333486269541368899, −5.48018582947015566709557836468, −5.41332074997876449372655765172, −5.31736567316366006737163490580, −4.83833496749622333183484247300, −4.69769013638679664970932068941, −4.34285810228409136575864701395, −4.16724879440638273612119743334, −3.88018655543076616919585141878, −3.82104548412570216815348924821, −3.54862701797093264924089218799, −3.50321927224742241712160031331, −2.95075291091458630666158686951, −2.46940522189583003796472085460, −2.31419947647168761468671289191, −2.11134169550118678652939388038, −2.06093373067615968602501284860, −1.79813658713328239981912201815, −0.76156251400893466076905475411, −0.74227996716052301592792367413, −0.64502171130161520560066485090,
0.64502171130161520560066485090, 0.74227996716052301592792367413, 0.76156251400893466076905475411, 1.79813658713328239981912201815, 2.06093373067615968602501284860, 2.11134169550118678652939388038, 2.31419947647168761468671289191, 2.46940522189583003796472085460, 2.95075291091458630666158686951, 3.50321927224742241712160031331, 3.54862701797093264924089218799, 3.82104548412570216815348924821, 3.88018655543076616919585141878, 4.16724879440638273612119743334, 4.34285810228409136575864701395, 4.69769013638679664970932068941, 4.83833496749622333183484247300, 5.31736567316366006737163490580, 5.41332074997876449372655765172, 5.48018582947015566709557836468, 5.87929871135333486269541368899, 6.00008704750756131241545925467, 6.21905585924481829084851771926, 6.34421449466675778362521060961, 6.56809349723715479604337229923
