L(s) = 1 | + 8·7-s + 4·17-s + 8·23-s + 6·25-s − 12·41-s + 16·47-s + 34·49-s + 24·71-s − 28·73-s − 16·79-s − 4·89-s − 4·97-s + 8·103-s − 4·113-s + 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯ |
L(s) = 1 | + 3.02·7-s + 0.970·17-s + 1.66·23-s + 6/5·25-s − 1.87·41-s + 2.33·47-s + 34/7·49-s + 2.84·71-s − 3.27·73-s − 1.80·79-s − 0.423·89-s − 0.406·97-s + 0.788·103-s − 0.376·113-s + 2.93·119-s + 1.63·121-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 + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.031968381\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.031968381\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.793760956749197758015793851178, −8.782890792057012148717923048519, −8.509755686003725234299752697398, −8.117736686076816284090835377516, −7.64161367339961920027084258393, −7.29963258870782483710472540334, −7.12309300166512313565346938203, −6.62180298459201982165834510160, −5.84930303477066848876127724748, −5.48692317258635396260497518279, −5.23660894146425618050434861345, −4.85617985742319696959468116782, −4.41324315214423738789174059540, −4.24199522480338224179017852798, −3.32705409726734369332605831933, −3.03417653043187040154959811955, −2.28141504243485475984353723068, −1.81598920577219138278859064481, −1.22108427632985252910194514225, −0.923414068528069850584061886816,
0.923414068528069850584061886816, 1.22108427632985252910194514225, 1.81598920577219138278859064481, 2.28141504243485475984353723068, 3.03417653043187040154959811955, 3.32705409726734369332605831933, 4.24199522480338224179017852798, 4.41324315214423738789174059540, 4.85617985742319696959468116782, 5.23660894146425618050434861345, 5.48692317258635396260497518279, 5.84930303477066848876127724748, 6.62180298459201982165834510160, 7.12309300166512313565346938203, 7.29963258870782483710472540334, 7.64161367339961920027084258393, 8.117736686076816284090835377516, 8.509755686003725234299752697398, 8.782890792057012148717923048519, 8.793760956749197758015793851178