| L(s) = 1 | − 7-s + 2·13-s + 3·19-s + 25-s − 3·31-s + 37-s − 2·61-s + 3·67-s − 73-s − 3·79-s − 2·91-s − 4·97-s + 3·103-s − 109-s − 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | − 7-s + 2·13-s + 3·19-s + 25-s − 3·31-s + 37-s − 2·61-s + 3·67-s − 73-s − 3·79-s − 2·91-s − 4·97-s + 3·103-s − 109-s − 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.028397683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028397683\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−10.22996797482655364332287354211, −9.943164268862807875968095886512, −9.441866102141382422424133166892, −9.213959693212811832901045368249, −8.838940813472546015952107453341, −8.399741372738161205283802021324, −7.71941402892878545355882274792, −7.52917761798086390072529399693, −6.89292234167930793368298031164, −6.74660570911995325271643102217, −5.90656580584473475784357339622, −5.77225174810134724728055507973, −5.35643391439854627798698363140, −4.77723423080864028136684722525, −3.81218958854971985760883422615, −3.78503175262904335487523784256, −3.05808760714726225457124507477, −2.87279342299094172998720509360, −1.60865109116096409236982239783, −1.12867824155953033217823816685,
1.12867824155953033217823816685, 1.60865109116096409236982239783, 2.87279342299094172998720509360, 3.05808760714726225457124507477, 3.78503175262904335487523784256, 3.81218958854971985760883422615, 4.77723423080864028136684722525, 5.35643391439854627798698363140, 5.77225174810134724728055507973, 5.90656580584473475784357339622, 6.74660570911995325271643102217, 6.89292234167930793368298031164, 7.52917761798086390072529399693, 7.71941402892878545355882274792, 8.399741372738161205283802021324, 8.838940813472546015952107453341, 9.213959693212811832901045368249, 9.441866102141382422424133166892, 9.943164268862807875968095886512, 10.22996797482655364332287354211
