| L(s) = 1 | − 1.73e3·25-s − 9.59e3·49-s − 3.26e4·73-s + 6.91e4·97-s + 1.82e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.14e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2.77·25-s − 3.99·49-s − 6.12·73-s + 7.34·97-s + 1.24·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 4·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + 1.94e−5·227-s + 1.90e−5·229-s + 1.84e−5·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.937932407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937932407\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( ( 1 + 866 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{4} T^{2} )^{2}( 1 + 2 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 9118 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 745438 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 478 T + p^{4} T^{2} )^{2}( 1 + 478 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 5425438 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 22852322 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8158 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 9118 T + p^{4} T^{2} )^{2}( 1 + 9118 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 77460958 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17282 T + p^{4} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.40911691012098779595899420480, −6.11718141334504651842939498480, −5.91502316207618303229255797189, −5.85682240623456399495117690071, −5.78127078206332020358062315721, −5.22709345542406298044276598320, −4.97780674230005250625358976915, −4.93804474889843855367366412971, −4.48567691961317954500348714526, −4.46689000271086164968067768927, −4.17722377595701834792411790034, −3.92877297904834336952540860670, −3.50108880248168702700496848955, −3.30664619927817130533561925273, −3.27603794112839908261394043996, −2.88604166443261535783560543421, −2.67476399869569389734934374283, −2.12343160604637611102546811650, −1.87926454805125467079732430759, −1.85743170420668400452501747101, −1.55718901603903829531114264876, −1.19886194255219767465124236808, −0.75295188812378473853297692815, −0.32988067863523977196776625835, −0.23551238621589921052438810502,
0.23551238621589921052438810502, 0.32988067863523977196776625835, 0.75295188812378473853297692815, 1.19886194255219767465124236808, 1.55718901603903829531114264876, 1.85743170420668400452501747101, 1.87926454805125467079732430759, 2.12343160604637611102546811650, 2.67476399869569389734934374283, 2.88604166443261535783560543421, 3.27603794112839908261394043996, 3.30664619927817130533561925273, 3.50108880248168702700496848955, 3.92877297904834336952540860670, 4.17722377595701834792411790034, 4.46689000271086164968067768927, 4.48567691961317954500348714526, 4.93804474889843855367366412971, 4.97780674230005250625358976915, 5.22709345542406298044276598320, 5.78127078206332020358062315721, 5.85682240623456399495117690071, 5.91502316207618303229255797189, 6.11718141334504651842939498480, 6.40911691012098779595899420480
