| L(s) = 1 | − 6·9-s − 2·11-s − 8·25-s − 8·29-s + 16·43-s − 12·53-s + 8·67-s + 24·71-s + 24·79-s + 27·81-s + 12·99-s − 8·109-s − 20·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·9-s − 0.603·11-s − 8/5·25-s − 1.48·29-s + 2.43·43-s − 1.64·53-s + 0.977·67-s + 2.84·71-s + 2.70·79-s + 3·81-s + 1.20·99-s − 0.766·109-s − 1.88·113-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836077418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836077418\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.982870676322914666258147112812, −7.81641009404354391744134080871, −7.29450663257727973833320704485, −6.94237552179863347379571789657, −6.45306181284646757897882998227, −6.17015629153820786769105593286, −5.84248296529944592015122544326, −5.53616967459082101241042101320, −5.23710442431232331758008912919, −5.06037535528847176776594851342, −4.29599160630098891137334237950, −4.10985400737049574255360821776, −3.47416686603466444580856063233, −3.40735689966014009037137882642, −2.82484703394178380554793926802, −2.39118950754729471895055389250, −2.10130451330242877490616940628, −1.71148395090277248856416209999, −0.56442379601835038706137805515, −0.53332021707265700610164652776,
0.53332021707265700610164652776, 0.56442379601835038706137805515, 1.71148395090277248856416209999, 2.10130451330242877490616940628, 2.39118950754729471895055389250, 2.82484703394178380554793926802, 3.40735689966014009037137882642, 3.47416686603466444580856063233, 4.10985400737049574255360821776, 4.29599160630098891137334237950, 5.06037535528847176776594851342, 5.23710442431232331758008912919, 5.53616967459082101241042101320, 5.84248296529944592015122544326, 6.17015629153820786769105593286, 6.45306181284646757897882998227, 6.94237552179863347379571789657, 7.29450663257727973833320704485, 7.81641009404354391744134080871, 7.982870676322914666258147112812
