| L(s) = 1 | − 4·7-s − 6·9-s − 4·25-s − 2·49-s + 24·63-s − 40·79-s + 27·81-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 16·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 2·9-s − 4/5·25-s − 2/7·49-s + 3.02·63-s − 4.50·79-s + 3·81-s + 1.81·121-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 + 1.20·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5925530762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5925530762\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.45515519822191585617861638758, −7.41694071745338381504248610325, −7.18924088271140654301828127326, −6.67228420866595203106852184293, −6.60429956932702705885237133666, −6.32208343310349744768570590630, −6.28163447997644455455584259464, −5.70988531001012266044390236368, −5.67082149846391749652466834147, −5.55878209027610727649949696689, −5.47761994803344308810944342912, −4.80111988350531236240150447404, −4.53851117458343401867218624076, −4.44904382789505592659387527999, −4.14857666032439485655813246039, −3.56616294633269589935088169911, −3.37042060032065511476602263885, −3.29502723690249553425052151119, −2.97395554584487724886649484246, −2.70062559873554309310896366729, −2.37420911996708029517998665191, −1.88121484176750506394937402220, −1.64788377480078058062671648637, −0.77161237921251292897969993807, −0.27933571258893938456168005239,
0.27933571258893938456168005239, 0.77161237921251292897969993807, 1.64788377480078058062671648637, 1.88121484176750506394937402220, 2.37420911996708029517998665191, 2.70062559873554309310896366729, 2.97395554584487724886649484246, 3.29502723690249553425052151119, 3.37042060032065511476602263885, 3.56616294633269589935088169911, 4.14857666032439485655813246039, 4.44904382789505592659387527999, 4.53851117458343401867218624076, 4.80111988350531236240150447404, 5.47761994803344308810944342912, 5.55878209027610727649949696689, 5.67082149846391749652466834147, 5.70988531001012266044390236368, 6.28163447997644455455584259464, 6.32208343310349744768570590630, 6.60429956932702705885237133666, 6.67228420866595203106852184293, 7.18924088271140654301828127326, 7.41694071745338381504248610325, 7.45515519822191585617861638758
