| L(s) = 1 | + 3-s + 13-s − 3·23-s + 25-s − 27-s + 39-s + 43-s − 49-s − 61-s − 3·69-s + 75-s − 79-s − 81-s + 3·101-s − 2·103-s − 3·113-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3-s + 13-s − 3·23-s + 25-s − 27-s + 39-s + 43-s − 49-s − 61-s − 3·69-s + 75-s − 79-s − 81-s + 3·101-s − 2·103-s − 3·113-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9717452144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9717452144\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.39765482749291717435591424634, −11.08984742979293985466481756946, −10.30831219857109700453183512117, −10.30506421165042085745412418728, −9.581638132499698438229462506422, −9.132386229608013730334562518720, −8.835815439870119628404488699575, −8.239370792711384438718748701436, −7.909557244304326782056395650432, −7.70590458351967196749016691747, −6.82056789875047686956044110771, −6.41015353755032203205142737256, −5.80927552704166062874431039270, −5.54327250578218456416062777967, −4.53489793647223695022936895715, −4.11911466133080515295842289531, −3.55865472456380189383507195910, −2.98529184696859381579188955908, −2.26548627772855935860652841815, −1.57386261404296858455354497471,
1.57386261404296858455354497471, 2.26548627772855935860652841815, 2.98529184696859381579188955908, 3.55865472456380189383507195910, 4.11911466133080515295842289531, 4.53489793647223695022936895715, 5.54327250578218456416062777967, 5.80927552704166062874431039270, 6.41015353755032203205142737256, 6.82056789875047686956044110771, 7.70590458351967196749016691747, 7.909557244304326782056395650432, 8.239370792711384438718748701436, 8.835815439870119628404488699575, 9.132386229608013730334562518720, 9.581638132499698438229462506422, 10.30506421165042085745412418728, 10.30831219857109700453183512117, 11.08984742979293985466481756946, 11.39765482749291717435591424634
