| L(s) = 1 | − 54·9-s + 500·25-s + 572·49-s + 4.76e3·73-s + 2.18e3·81-s − 5.32e3·97-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.01e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.70e4·225-s + ⋯ |
| L(s) = 1 | − 2·9-s + 4·25-s + 1.66·49-s + 7.63·73-s + 3·81-s − 5.56·97-s − 4·121-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 − 0.460·169-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 − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.387537351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.387537351\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 286 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 506 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2}( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 35282 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 89206 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )^{2}( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 420838 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )^{2}( 1 + 880 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 204622 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{4} \) |
| 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
−6.89616658590324134982440389039, −6.64952778357427047224078442579, −6.64580594916964061611983978638, −6.32031862625315723075361792527, −6.24939667281495844412588984608, −5.73707978889462769723921989018, −5.47195390308367800905363502166, −5.30494943420950199226690934555, −5.11380273010755289962214744801, −5.03895185988192245784834533503, −4.77826379520547456650558584187, −4.34566979332851233265595937156, −3.85127149546011819197941699583, −3.81548242018842548565866858009, −3.66632269022715294249112236721, −3.09628983687341608600615195888, −2.85774696676421061857680966095, −2.72912180168898248382655574491, −2.38341297517802458351914181864, −2.37195631231174284133981302331, −1.61495955298535650365018045490, −1.33304816161354853340410400382, −0.843982908741623644452368033030, −0.57274434585369011270390644167, −0.41878196949944145553629515275,
0.41878196949944145553629515275, 0.57274434585369011270390644167, 0.843982908741623644452368033030, 1.33304816161354853340410400382, 1.61495955298535650365018045490, 2.37195631231174284133981302331, 2.38341297517802458351914181864, 2.72912180168898248382655574491, 2.85774696676421061857680966095, 3.09628983687341608600615195888, 3.66632269022715294249112236721, 3.81548242018842548565866858009, 3.85127149546011819197941699583, 4.34566979332851233265595937156, 4.77826379520547456650558584187, 5.03895185988192245784834533503, 5.11380273010755289962214744801, 5.30494943420950199226690934555, 5.47195390308367800905363502166, 5.73707978889462769723921989018, 6.24939667281495844412588984608, 6.32031862625315723075361792527, 6.64580594916964061611983978638, 6.64952778357427047224078442579, 6.89616658590324134982440389039
