| L(s) = 1 | + 64·9-s + 728·25-s − 896·29-s + 208·53-s + 1.40e3·81-s + 2.46e3·109-s − 8.64e3·113-s + 6.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.00e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2.37·9-s + 5.82·25-s − 5.73·29-s + 0.539·53-s + 1.93·81-s + 2.16·109-s − 7.19·113-s + 5.01·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.18·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(35.70082199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.70082199\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - 32 T^{2} + 832 T^{4} - 32 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 5 | \( ( 1 - 364 T^{2} + 63982 T^{4} - 364 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3336 T^{2} + 5345346 T^{4} - 3336 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 3500 T^{2} + 6393550 T^{4} - 3500 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 11760 T^{2} + 81461760 T^{4} - 11760 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 + 15984 T^{2} + 157003328 T^{4} + 15984 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 + 12540 T^{2} + 295243878 T^{4} + 12540 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 224 T + 60930 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 55324 T^{2} + 1521564358 T^{4} + 55324 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 + 24474 T^{2} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 124768 T^{2} + 13390720576 T^{4} - 124768 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 55800 T^{2} + 13284680898 T^{4} + 55800 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 223436 T^{2} + 28034138950 T^{4} + 223436 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 52 T + 185142 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 + 548992 T^{2} + 149277953376 T^{4} + 548992 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 872844 T^{2} + 293483629518 T^{4} - 872844 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 955868 T^{2} + 401042952694 T^{4} - 955868 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 992156 T^{2} + 502163893734 T^{4} - 992156 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 769664 T^{2} + 312561795520 T^{4} - 769664 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 569132 T^{2} + 298725143590 T^{4} - 569132 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1643680 T^{2} + 1285997253120 T^{4} + 1643680 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 2816016 T^{2} + 2976448615968 T^{4} - 2816016 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 765408 T^{2} + 527893590912 T^{4} - 765408 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 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
−3.99873730906402896069630773402, −3.91021696746004479386182661227, −3.78530249510133930228166582496, −3.57761050004487995993323178080, −3.46391820393132824213039881827, −3.42308620155341407034784771616, −3.15047351761297301864910652534, −2.92427759923097460019657038917, −2.91270532006465184447051615818, −2.75384699794676937459383217588, −2.71639876453844656605117014562, −2.51479501340883220471885827317, −2.09623641115325548763520681422, −2.06289762262752062557323513570, −1.82512391100654682589272542613, −1.75188923053561740378985521806, −1.56647334116511102702955655867, −1.54017407889677632125207839486, −1.44783738433954088060115158626, −0.993859496749190770960423728518, −0.928804277201356343153907793326, −0.67397130468330924285860870673, −0.53867382928854658716564209030, −0.36709524361443185786620215775, −0.36217215043425742557695822885,
0.36217215043425742557695822885, 0.36709524361443185786620215775, 0.53867382928854658716564209030, 0.67397130468330924285860870673, 0.928804277201356343153907793326, 0.993859496749190770960423728518, 1.44783738433954088060115158626, 1.54017407889677632125207839486, 1.56647334116511102702955655867, 1.75188923053561740378985521806, 1.82512391100654682589272542613, 2.06289762262752062557323513570, 2.09623641115325548763520681422, 2.51479501340883220471885827317, 2.71639876453844656605117014562, 2.75384699794676937459383217588, 2.91270532006465184447051615818, 2.92427759923097460019657038917, 3.15047351761297301864910652534, 3.42308620155341407034784771616, 3.46391820393132824213039881827, 3.57761050004487995993323178080, 3.78530249510133930228166582496, 3.91021696746004479386182661227, 3.99873730906402896069630773402
Plot not available for L-functions of degree greater than 10.
