L(s) = 1 | − 72·11-s + 7·16-s + 88·31-s − 72·41-s + 8·61-s + 288·71-s − 9·81-s − 432·101-s + 2.75e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 504·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 6.54·11-s + 7/16·16-s + 2.83·31-s − 1.75·41-s + 8/61·61-s + 4.05·71-s − 1/9·81-s − 4.27·101-s + 22.7·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s − 2.86·176-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5464015786\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5464015786\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 4702 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 113858 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 475582 T^{4} + p^{8} T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 3168578 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 2956702 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 9478462 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 15243938 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 14394142 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 10963582 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 7582 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 31395742 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 7742 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 171536062 T^{4} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43272677271919347671140114634, −10.32854973849335610960309268242, −10.08845407323857217880959204922, −9.739890412006283037553083325122, −9.737129714682633257585266768364, −8.941822631590589330935127162669, −8.297992431837071455606485224541, −8.242542159109238893010863315701, −8.196381258044041905517853924566, −7.84828196928130944765311894468, −7.66229350958728451736624291298, −7.27028704252744003276067801359, −6.64449596392523289051848591996, −6.54992475316093479185306346998, −5.71175058250511466770320981869, −5.53765216867931320478999536059, −5.12847651918852031711671823561, −5.11961660639705437874096027671, −4.76273541405197502558615016847, −4.10780310830882473457413560765, −3.10813538946266115431392591894, −3.00536279826085397227320469675, −2.47970882258389291612008452390, −2.26277832427149618763724769302, −0.40958377309226192183101428220,
0.40958377309226192183101428220, 2.26277832427149618763724769302, 2.47970882258389291612008452390, 3.00536279826085397227320469675, 3.10813538946266115431392591894, 4.10780310830882473457413560765, 4.76273541405197502558615016847, 5.11961660639705437874096027671, 5.12847651918852031711671823561, 5.53765216867931320478999536059, 5.71175058250511466770320981869, 6.54992475316093479185306346998, 6.64449596392523289051848591996, 7.27028704252744003276067801359, 7.66229350958728451736624291298, 7.84828196928130944765311894468, 8.196381258044041905517853924566, 8.242542159109238893010863315701, 8.297992431837071455606485224541, 8.941822631590589330935127162669, 9.737129714682633257585266768364, 9.739890412006283037553083325122, 10.08845407323857217880959204922, 10.32854973849335610960309268242, 10.43272677271919347671140114634