| L(s) = 1 | − 8·7-s − 8·25-s − 32·37-s + 16·43-s + 28·49-s − 32·67-s + 64·79-s + 32·109-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 64·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 8/5·25-s − 5.26·37-s + 2.43·43-s + 4·49-s − 3.90·67-s + 7.20·79-s + 3.06·109-s + 5.81·121-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 + 3.07·169-s + 0.0760·173-s + 4.83·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4532473785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4532473785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 12 T^{2} + 582 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T^{2} + 1078 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 48 T^{2} + 1346 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 80 T^{2} + 3154 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T^{2} - 122 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 84 T^{2} + 3558 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 176 T^{2} + 13234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 208 T^{2} + 20610 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 132 T^{2} + 8742 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 276 T^{2} + 34854 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 228 T^{2} + 31686 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.55570125385755950123583228550, −3.34567528949381492016485147120, −3.17115358202344242756802451574, −3.16735598379207830099146509514, −3.09038328900078169910217317804, −2.99702697153643017818909711131, −2.90817570446132317028905865822, −2.70744982575330537515706450217, −2.47932400773668284890602011379, −2.41898796968907885261094171356, −2.27504958246501217144529474002, −2.23993997800414770018967184748, −2.07098173943504441442458408217, −1.91019926360896291257226374296, −1.71940199412349086901178426883, −1.65723753877153267290185306873, −1.62488931471116876300727004175, −1.56284705793649775867005365036, −1.14999303700002250799834961115, −0.870961302042309793377239536467, −0.76160539689810389705420766129, −0.63869437015055235086377417755, −0.58816187871477779800986081694, −0.24264439834399897995944643148, −0.093068359409457587351585262317,
0.093068359409457587351585262317, 0.24264439834399897995944643148, 0.58816187871477779800986081694, 0.63869437015055235086377417755, 0.76160539689810389705420766129, 0.870961302042309793377239536467, 1.14999303700002250799834961115, 1.56284705793649775867005365036, 1.62488931471116876300727004175, 1.65723753877153267290185306873, 1.71940199412349086901178426883, 1.91019926360896291257226374296, 2.07098173943504441442458408217, 2.23993997800414770018967184748, 2.27504958246501217144529474002, 2.41898796968907885261094171356, 2.47932400773668284890602011379, 2.70744982575330537515706450217, 2.90817570446132317028905865822, 2.99702697153643017818909711131, 3.09038328900078169910217317804, 3.16735598379207830099146509514, 3.17115358202344242756802451574, 3.34567528949381492016485147120, 3.55570125385755950123583228550
Plot not available for L-functions of degree greater than 10.
