L(s) = 1 | + 4·4-s + 12·16-s − 12·19-s + 12·31-s + 26·49-s − 24·61-s + 48·64-s − 48·76-s + 20·79-s − 4·109-s − 40·121-s + 48·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 2·4-s + 3·16-s − 2.75·19-s + 2.15·31-s + 26/7·49-s − 3.07·61-s + 6·64-s − 5.50·76-s + 2.25·79-s − 0.383·109-s − 3.63·121-s + 4.31·124-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 + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.551244569\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551244569\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
good | 2 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 10 T^{2} - 189 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} - 333 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 56 T^{2} + 927 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 98 T^{2} + 6795 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | \( ( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.06982155015789405412873777398, −3.92157843483627024840577532895, −3.62541384226967002448581676508, −3.61962244900445663104858370809, −3.43368533304455412826007418105, −3.22427942568721338484209804262, −3.16922010203841444705619912149, −3.14138160525375503093535285156, −3.09190002670332119830228443292, −2.66321864480005228017365175026, −2.57375856413937178662158169198, −2.49741836211945352553979329941, −2.37370319368140202659082663126, −2.28027526056651977732342756408, −2.09521962555785351838246579840, −2.08973064326958645177339882740, −1.95249731965899687470009127079, −1.82861426733314264431994659458, −1.32535572009225789476010591542, −1.27959434471471848582613754179, −1.09044688085996643423897744174, −1.00025380112030553367748671497, −0.954812901784168572942617613920, −0.43986865154136846764623919266, −0.12225798077449202764645261149,
0.12225798077449202764645261149, 0.43986865154136846764623919266, 0.954812901784168572942617613920, 1.00025380112030553367748671497, 1.09044688085996643423897744174, 1.27959434471471848582613754179, 1.32535572009225789476010591542, 1.82861426733314264431994659458, 1.95249731965899687470009127079, 2.08973064326958645177339882740, 2.09521962555785351838246579840, 2.28027526056651977732342756408, 2.37370319368140202659082663126, 2.49741836211945352553979329941, 2.57375856413937178662158169198, 2.66321864480005228017365175026, 3.09190002670332119830228443292, 3.14138160525375503093535285156, 3.16922010203841444705619912149, 3.22427942568721338484209804262, 3.43368533304455412826007418105, 3.61962244900445663104858370809, 3.62541384226967002448581676508, 3.92157843483627024840577532895, 4.06982155015789405412873777398
Plot not available for L-functions of degree greater than 10.