| L(s) = 1 | + 16·7-s + 28·17-s − 304·23-s + 138·25-s − 448·31-s + 140·41-s + 672·47-s − 494·49-s − 144·71-s − 588·73-s + 928·79-s − 532·89-s + 1.98e3·97-s − 2.35e3·103-s + 3.42e3·113-s + 448·119-s + 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.86e3·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.863·7-s + 0.399·17-s − 2.75·23-s + 1.10·25-s − 2.59·31-s + 0.533·41-s + 2.08·47-s − 1.44·49-s − 0.240·71-s − 0.942·73-s + 1.32·79-s − 0.633·89-s + 2.08·97-s − 2.24·103-s + 2.84·113-s + 0.345·119-s + 1.81·121-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 − 2.38·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.087481135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087481135\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 138 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 994 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.50905026458588040193845583089, −11.08809667607036098817194099997, −10.83158076103272159622626065689, −10.11595206648191875139813179680, −9.882038112594304649892676144748, −9.132515518959106249642898717489, −8.799322300174788621536040335678, −8.198969339855747519934614788121, −7.72570470739018124956570362382, −7.39109806090272817479364518718, −6.76704944160135269780785338037, −5.84985551904905678596625356553, −5.80721919713589773528495381596, −5.00570808115154024738561017517, −4.37683703969393919930731561558, −3.84827417093282664065670377297, −3.16123559013333408038601033295, −2.09989410743420151210019375166, −1.71875524449671354734302091813, −0.54128679824374338364716470310,
0.54128679824374338364716470310, 1.71875524449671354734302091813, 2.09989410743420151210019375166, 3.16123559013333408038601033295, 3.84827417093282664065670377297, 4.37683703969393919930731561558, 5.00570808115154024738561017517, 5.80721919713589773528495381596, 5.84985551904905678596625356553, 6.76704944160135269780785338037, 7.39109806090272817479364518718, 7.72570470739018124956570362382, 8.198969339855747519934614788121, 8.799322300174788621536040335678, 9.132515518959106249642898717489, 9.882038112594304649892676144748, 10.11595206648191875139813179680, 10.83158076103272159622626065689, 11.08809667607036098817194099997, 11.50905026458588040193845583089
