| L(s) = 1 | + 2·9-s − 12·17-s + 2·25-s − 18·49-s − 56·73-s − 15·81-s + 40·89-s − 8·97-s − 56·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.91·17-s + 2/5·25-s − 2.57·49-s − 6.55·73-s − 5/3·81-s + 4.23·89-s − 0.812·97-s − 5.26·113-s + 4/11·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.755716741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.755716741\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−6.10746565822432930091051026768, −5.94253520243105416126131206293, −5.86379107399020928099026297478, −5.45387803173768776976184704064, −5.23116982894672747637051606040, −5.05303213584734202699689688518, −4.93670705825449601930891939898, −4.47870854996578677207664046055, −4.39125785254593678461574087875, −4.33468644254046648056491850628, −4.31850489629127107782029398352, −3.93073584593977016905232276533, −3.59811192331837045214888222113, −3.35588060359060981566985045467, −3.13000360775533887724957644628, −2.85817313257320118487583343621, −2.63571475388658294722741457533, −2.60814400776212100639895005951, −2.00428988594600100214551510481, −1.92462084059525295215857969172, −1.65138137018106984412920096306, −1.41432921684726186387629782177, −1.16725832193935022155067138059, −0.36781695494391838827899421765, −0.34352469053775353781739827941,
0.34352469053775353781739827941, 0.36781695494391838827899421765, 1.16725832193935022155067138059, 1.41432921684726186387629782177, 1.65138137018106984412920096306, 1.92462084059525295215857969172, 2.00428988594600100214551510481, 2.60814400776212100639895005951, 2.63571475388658294722741457533, 2.85817313257320118487583343621, 3.13000360775533887724957644628, 3.35588060359060981566985045467, 3.59811192331837045214888222113, 3.93073584593977016905232276533, 4.31850489629127107782029398352, 4.33468644254046648056491850628, 4.39125785254593678461574087875, 4.47870854996578677207664046055, 4.93670705825449601930891939898, 5.05303213584734202699689688518, 5.23116982894672747637051606040, 5.45387803173768776976184704064, 5.86379107399020928099026297478, 5.94253520243105416126131206293, 6.10746565822432930091051026768
