| L(s) = 1 | + 8·7-s + 4·17-s − 16·23-s + 10·25-s − 8·31-s + 12·41-s + 16·47-s + 34·49-s + 16·71-s + 12·73-s − 8·79-s − 12·89-s − 4·97-s − 8·103-s + 28·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 0.970·17-s − 3.33·23-s + 2·25-s − 1.43·31-s + 1.87·41-s + 2.33·47-s + 34/7·49-s + 1.89·71-s + 1.40·73-s − 0.900·79-s − 1.27·89-s − 0.406·97-s − 0.788·103-s + 2.63·113-s + 2.93·119-s + 6/11·121-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 − 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.540326112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.540326112\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.890884584524078086460046111068, −9.757819236639717806212717933968, −9.016393465530618757925132762868, −8.650811057674037010870100150293, −8.295967730300905786661193294856, −7.966622569732101710547939492039, −7.63455940501931738165346753153, −7.35659034201045942356868869490, −6.81030811603774714024506636250, −5.97246099724709468743033533834, −5.64346789004486508333665991381, −5.43998307798370986961544031869, −4.72385712682161959854602407232, −4.48673801811247150812678591755, −3.98951474176203430726040510357, −3.52199049335282005802690539379, −2.33665932454297985572256525507, −2.24257550388304404591405537177, −1.48666305912036011624599909232, −0.911501807918419972467098233395,
0.911501807918419972467098233395, 1.48666305912036011624599909232, 2.24257550388304404591405537177, 2.33665932454297985572256525507, 3.52199049335282005802690539379, 3.98951474176203430726040510357, 4.48673801811247150812678591755, 4.72385712682161959854602407232, 5.43998307798370986961544031869, 5.64346789004486508333665991381, 5.97246099724709468743033533834, 6.81030811603774714024506636250, 7.35659034201045942356868869490, 7.63455940501931738165346753153, 7.966622569732101710547939492039, 8.295967730300905786661193294856, 8.650811057674037010870100150293, 9.016393465530618757925132762868, 9.757819236639717806212717933968, 9.890884584524078086460046111068
