| L(s) = 1 | + 8·9-s + 48·41-s + 8·49-s + 30·81-s − 24·89-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-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 | + 8/3·9-s + 7.49·41-s + 8/7·49-s + 10/3·81-s − 2.54·89-s − 4·121-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 − 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.329796544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.329796544\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−5.70136602064571688255490547342, −5.39626410643701680754440301210, −5.33788310706385739388702613838, −4.97472278276881156017288804561, −4.94921603424922444341273411205, −4.50917411383501438840696031707, −4.39293310433670662386798343917, −4.25985471766429859919869643486, −4.18551643082977266123687688909, −4.05169645381293215502582831559, −3.79900489810019940404714660244, −3.67152415125918549243520950774, −3.43453838122108151308317996457, −2.85006167458386291820353692678, −2.83921206954150085600614730334, −2.57952931828557483201308391035, −2.54250479292172834031722608755, −2.21372874559264156248051810648, −2.01955599413479881017551878874, −1.49352982124345944233383597370, −1.45030015060695382885101537357, −1.22495267295920538543140061602, −0.894139317004329303341185546645, −0.75265696496809349060463928416, −0.35030165214176267515189230727,
0.35030165214176267515189230727, 0.75265696496809349060463928416, 0.894139317004329303341185546645, 1.22495267295920538543140061602, 1.45030015060695382885101537357, 1.49352982124345944233383597370, 2.01955599413479881017551878874, 2.21372874559264156248051810648, 2.54250479292172834031722608755, 2.57952931828557483201308391035, 2.83921206954150085600614730334, 2.85006167458386291820353692678, 3.43453838122108151308317996457, 3.67152415125918549243520950774, 3.79900489810019940404714660244, 4.05169645381293215502582831559, 4.18551643082977266123687688909, 4.25985471766429859919869643486, 4.39293310433670662386798343917, 4.50917411383501438840696031707, 4.94921603424922444341273411205, 4.97472278276881156017288804561, 5.33788310706385739388702613838, 5.39626410643701680754440301210, 5.70136602064571688255490547342
