| L(s) = 1 | − 12·11-s − 8·23-s − 2·25-s + 12·29-s + 4·37-s + 20·43-s + 4·53-s + 8·67-s + 24·71-s − 8·79-s − 8·107-s − 20·109-s − 8·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.61·11-s − 1.66·23-s − 2/5·25-s + 2.22·29-s + 0.657·37-s + 3.04·43-s + 0.549·53-s + 0.977·67-s + 2.84·71-s − 0.900·79-s − 0.773·107-s − 1.91·109-s − 0.752·113-s + 7.81·121-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 + 6/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238197573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238197573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.417785025089098975517243289351, −8.282450084023836002208102668313, −8.022973242666175078235766606173, −7.82623837176705370339088250099, −7.37019172373713341674979438395, −7.05048551759444525519261316964, −6.42702747481272562963285536877, −6.10075081561229970736180324067, −5.63252449148214209221580864945, −5.40917985134451442849753675087, −5.04024085589718567933654509490, −4.62675643845550271138437381653, −4.11528239003647898410515933592, −3.82948899268086455961308645554, −2.85799735933844640095495338182, −2.83672231993995609620517675329, −2.33245980035607948818486635342, −2.08263817056232641145314108233, −0.948427762321686055952858963498, −0.38859476264435154329064331849,
0.38859476264435154329064331849, 0.948427762321686055952858963498, 2.08263817056232641145314108233, 2.33245980035607948818486635342, 2.83672231993995609620517675329, 2.85799735933844640095495338182, 3.82948899268086455961308645554, 4.11528239003647898410515933592, 4.62675643845550271138437381653, 5.04024085589718567933654509490, 5.40917985134451442849753675087, 5.63252449148214209221580864945, 6.10075081561229970736180324067, 6.42702747481272562963285536877, 7.05048551759444525519261316964, 7.37019172373713341674979438395, 7.82623837176705370339088250099, 8.022973242666175078235766606173, 8.282450084023836002208102668313, 8.417785025089098975517243289351
