| L(s) = 1 | − 72·13-s + 64·25-s − 144·37-s − 52·49-s + 144·61-s + 224·73-s + 416·97-s − 504·109-s + 452·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.56e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.53·13-s + 2.55·25-s − 3.89·37-s − 1.06·49-s + 2.36·61-s + 3.06·73-s + 4.28·97-s − 4.62·109-s + 3.73·121-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 + 15.1·169-s + 0.00578·173-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.411082858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411082858\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 226 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 434 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1850 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 2480 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 910 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 880 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5134 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11834 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 8002 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7904 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
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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
−6.47686210757362569061280899583, −5.84602921663563060761745103298, −5.69393668915231931226153486815, −5.37195483105497565893227966058, −5.24279078725241668525995237304, −5.09230066871313921717819478904, −4.97673044748954447066662998972, −4.89685206657229029684338896292, −4.68555042146248297495289192696, −4.35412482994673377403389677284, −4.21199786160443290383008243766, −3.76800123597624660521286335433, −3.63635990454129093624345137469, −3.14028042682206819849667513286, −3.09312896121628518259025096678, −2.88320748488929123990440337544, −2.73131648962374794051657063034, −2.22578019307554443463988936915, −2.03326848536155165884815230798, −1.97189490963984873577917121375, −1.89136409030559965547500338682, −1.12800774385316076061086065240, −0.71556219710755133327387816191, −0.48422164590159127965190985756, −0.21365290439948183879069952290,
0.21365290439948183879069952290, 0.48422164590159127965190985756, 0.71556219710755133327387816191, 1.12800774385316076061086065240, 1.89136409030559965547500338682, 1.97189490963984873577917121375, 2.03326848536155165884815230798, 2.22578019307554443463988936915, 2.73131648962374794051657063034, 2.88320748488929123990440337544, 3.09312896121628518259025096678, 3.14028042682206819849667513286, 3.63635990454129093624345137469, 3.76800123597624660521286335433, 4.21199786160443290383008243766, 4.35412482994673377403389677284, 4.68555042146248297495289192696, 4.89685206657229029684338896292, 4.97673044748954447066662998972, 5.09230066871313921717819478904, 5.24279078725241668525995237304, 5.37195483105497565893227966058, 5.69393668915231931226153486815, 5.84602921663563060761745103298, 6.47686210757362569061280899583
