| L(s) = 1 | + 16·19-s − 10·25-s − 4·31-s + 16·37-s + 32·103-s − 16·109-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.67·19-s − 2·25-s − 0.718·31-s + 2.63·37-s + 3.15·103-s − 1.53·109-s + 1.27·121-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 + 2.15·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.975605915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.975605915\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−5.60357944718570714092716278009, −5.51979078214773733092702489143, −5.42027437219794824652123258076, −4.85201633656257562332589149150, −4.80644378787604916477876619009, −4.68115378228741512986835231101, −4.61173007060716447557845031909, −4.15482889882680533426857102683, −4.10755386262384893302972959422, −3.82699095089392743476653733585, −3.68812297127066742060218331275, −3.26149578250211477175875619857, −3.23744140325021410495133196844, −3.20988357544793288953886458600, −3.03865359479527171503023370452, −2.43026732656221220935317623142, −2.35646533275950541110443070020, −2.30549641026596195091709438638, −2.02212308629211435461448464649, −1.53504394174330797523206353312, −1.42725942948287277683751612130, −1.13749648477910573252393607553, −0.908493481907067321896044956712, −0.63329011669784976241150922127, −0.24753622179303892227757987686,
0.24753622179303892227757987686, 0.63329011669784976241150922127, 0.908493481907067321896044956712, 1.13749648477910573252393607553, 1.42725942948287277683751612130, 1.53504394174330797523206353312, 2.02212308629211435461448464649, 2.30549641026596195091709438638, 2.35646533275950541110443070020, 2.43026732656221220935317623142, 3.03865359479527171503023370452, 3.20988357544793288953886458600, 3.23744140325021410495133196844, 3.26149578250211477175875619857, 3.68812297127066742060218331275, 3.82699095089392743476653733585, 4.10755386262384893302972959422, 4.15482889882680533426857102683, 4.61173007060716447557845031909, 4.68115378228741512986835231101, 4.80644378787604916477876619009, 4.85201633656257562332589149150, 5.42027437219794824652123258076, 5.51979078214773733092702489143, 5.60357944718570714092716278009
