L(s) = 1 | + 3·19-s − 25-s + 3·31-s − 37-s + 2·43-s + 67-s − 3·73-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 3·19-s − 25-s + 3·31-s − 37-s + 2·43-s + 67-s − 3·73-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.343485828\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.343485828\) |
\(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.687184356593749704493411338288, −9.275247870096338604079306453823, −9.081666323979300436048649978424, −8.403023092783124412015173049755, −7.972400335505062075757378460648, −7.891062340729175464212660120786, −7.18860862345448173196603323893, −7.11930821621898335104347898863, −6.49898595070475263277650220619, −6.02399397173807095499728119634, −5.53021260668522422814805144838, −5.42507612909214849578848355285, −4.62941671262291896795223466798, −4.49610482149762231237602058076, −3.70144622381404672599981968592, −3.41057191562855977404957052690, −2.61058847721582603051547490613, −2.60865273860038604852418654443, −1.39681928976638470793928278990, −1.04076825266703248020012614521,
1.04076825266703248020012614521, 1.39681928976638470793928278990, 2.60865273860038604852418654443, 2.61058847721582603051547490613, 3.41057191562855977404957052690, 3.70144622381404672599981968592, 4.49610482149762231237602058076, 4.62941671262291896795223466798, 5.42507612909214849578848355285, 5.53021260668522422814805144838, 6.02399397173807095499728119634, 6.49898595070475263277650220619, 7.11930821621898335104347898863, 7.18860862345448173196603323893, 7.891062340729175464212660120786, 7.972400335505062075757378460648, 8.403023092783124412015173049755, 9.081666323979300436048649978424, 9.275247870096338604079306453823, 9.687184356593749704493411338288