| L(s) = 1 | + 3-s + 11-s + 25-s − 27-s + 33-s − 3·41-s + 3·43-s + 49-s − 59-s − 3·67-s + 2·73-s + 75-s − 81-s − 2·83-s + 97-s + 2·107-s + 121-s − 3·123-s + 127-s + 3·129-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3-s + 11-s + 25-s − 27-s + 33-s − 3·41-s + 3·43-s + 49-s − 59-s − 3·67-s + 2·73-s + 75-s − 81-s − 2·83-s + 97-s + 2·107-s + 121-s − 3·123-s + 127-s + 3·129-s + 131-s + 137-s + 139-s + 147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.490359968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490359968\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−10.07443358411088344535432897769, −9.660693034100202583981091375850, −9.337354543190856321723673837020, −8.775020080361638186348592034790, −8.719966620729926208168790030478, −8.389269550096887448173989284572, −7.66951300868047803755641760938, −7.36376255195429644191716471930, −7.05928280792007312886934696096, −6.46729429166528441585294980128, −5.96271890973023822389230858360, −5.71830336706417071697583893626, −4.92844384708860131588241994054, −4.55014463992081048831084977891, −4.01774777797790538428691430007, −3.45136144260439441947776489926, −3.12486692057215219076220711185, −2.49722513095698334688082160395, −1.89784221832519258796381481516, −1.16313552105860819673581777060,
1.16313552105860819673581777060, 1.89784221832519258796381481516, 2.49722513095698334688082160395, 3.12486692057215219076220711185, 3.45136144260439441947776489926, 4.01774777797790538428691430007, 4.55014463992081048831084977891, 4.92844384708860131588241994054, 5.71830336706417071697583893626, 5.96271890973023822389230858360, 6.46729429166528441585294980128, 7.05928280792007312886934696096, 7.36376255195429644191716471930, 7.66951300868047803755641760938, 8.389269550096887448173989284572, 8.719966620729926208168790030478, 8.775020080361638186348592034790, 9.337354543190856321723673837020, 9.660693034100202583981091375850, 10.07443358411088344535432897769
