| L(s) = 1 | + 8·25-s + 32·37-s + 14·49-s + 40·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 8/5·25-s + 5.26·37-s + 2·49-s + 3.83·109-s − 3.63·121-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 + 4·169-s + 0.0760·173-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.524996644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.524996644\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 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
−5.95411403885119429529834320209, −5.76126058235073063800590026815, −5.74945051942869405790406041071, −5.20620337995300201556789124674, −5.20559501786000294310963643611, −5.08669013305805620823043701770, −4.69451597921007789672670211398, −4.50819023170234699227523999283, −4.31741667623746288891483386290, −4.30556155439045715125839943316, −3.92917414666715754348193004644, −3.85394391229596440689945421837, −3.54977944803037170831471222398, −3.23313189659204189205082721633, −2.89055737892493040800202725800, −2.77357761669477770010909330436, −2.74675619590804816107286643587, −2.46655462110264676192881976346, −2.12522913673909675416089481199, −1.85889090997514247765741724530, −1.63431168913808790160762836562, −1.02768015994618399583185250175, −0.960822308585700928598324417373, −0.77912361408718446470305494861, −0.39511703486023966124287967588,
0.39511703486023966124287967588, 0.77912361408718446470305494861, 0.960822308585700928598324417373, 1.02768015994618399583185250175, 1.63431168913808790160762836562, 1.85889090997514247765741724530, 2.12522913673909675416089481199, 2.46655462110264676192881976346, 2.74675619590804816107286643587, 2.77357761669477770010909330436, 2.89055737892493040800202725800, 3.23313189659204189205082721633, 3.54977944803037170831471222398, 3.85394391229596440689945421837, 3.92917414666715754348193004644, 4.30556155439045715125839943316, 4.31741667623746288891483386290, 4.50819023170234699227523999283, 4.69451597921007789672670211398, 5.08669013305805620823043701770, 5.20559501786000294310963643611, 5.20620337995300201556789124674, 5.74945051942869405790406041071, 5.76126058235073063800590026815, 5.95411403885119429529834320209
