L(s) = 1 | + 4·11-s − 4·17-s − 8·19-s − 8·25-s − 12·41-s − 16·43-s − 12·49-s + 24·59-s − 16·67-s − 16·73-s + 12·83-s − 4·89-s − 28·97-s + 8·107-s − 12·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 0.970·17-s − 1.83·19-s − 8/5·25-s − 1.87·41-s − 2.43·43-s − 1.71·49-s + 3.12·59-s − 1.95·67-s − 1.87·73-s + 1.31·83-s − 0.423·89-s − 2.84·97-s + 0.773·107-s − 1.12·113-s − 0.909·121-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 − 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.145324295612495077388710810711, −8.083400092509309447820919807206, −7.11174278129634174012226832658, −7.09380594818878033884864566768, −6.58216225485557033095722837256, −6.52817398546664431634549984838, −5.99076097101517058868882678531, −5.65796765002052045802713295943, −5.15977761347797516334687653687, −4.69744611970923338206884747787, −4.31680064464861127947847794584, −4.09271876452117206942159769554, −3.50878290635458755194079886524, −3.32264831977791976813729633832, −2.57891094390067338502961656788, −2.01235288403696179765377395125, −1.77329287335247602298611789780, −1.28028641527722770251757283248, 0, 0,
1.28028641527722770251757283248, 1.77329287335247602298611789780, 2.01235288403696179765377395125, 2.57891094390067338502961656788, 3.32264831977791976813729633832, 3.50878290635458755194079886524, 4.09271876452117206942159769554, 4.31680064464861127947847794584, 4.69744611970923338206884747787, 5.15977761347797516334687653687, 5.65796765002052045802713295943, 5.99076097101517058868882678531, 6.52817398546664431634549984838, 6.58216225485557033095722837256, 7.09380594818878033884864566768, 7.11174278129634174012226832658, 8.083400092509309447820919807206, 8.145324295612495077388710810711