| L(s) = 1 | − 3-s + 4-s − 12-s − 13-s − 3·19-s − 2·25-s + 27-s + 3·37-s + 39-s + 43-s − 52-s + 3·57-s + 61-s − 64-s + 2·75-s − 3·76-s − 4·79-s − 81-s + 3·97-s − 2·100-s + 2·103-s + 108-s − 3·111-s + 121-s + 127-s − 129-s + 131-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 12-s − 13-s − 3·19-s − 2·25-s + 27-s + 3·37-s + 39-s + 43-s − 52-s + 3·57-s + 61-s − 64-s + 2·75-s − 3·76-s − 4·79-s − 81-s + 3·97-s − 2·100-s + 2·103-s + 108-s − 3·111-s + 121-s + 127-s − 129-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6728927704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6728927704\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$ | \( ( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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
−9.797350943505261902503758501593, −9.078603658779419018443433569790, −8.949804852530515720052663925626, −8.226722828839490055123986141565, −8.097128653855452684735595781446, −7.38330118863900126867678063803, −7.36350503116856883740770103992, −6.64516734715479297145623861589, −6.39896875523237896851727197586, −5.92768647642378177697410623732, −5.91877674034631193595653990602, −5.32468806028025142523235153501, −4.51609018354352792829453870868, −4.32451662173143775521012180716, −4.14964012363205981019583132946, −3.09553092393455181817991695072, −2.66745555936155289403605490349, −2.00500247912293567011383727501, −2.00166582799035419287843852625, −0.59619517928034155510826749509,
0.59619517928034155510826749509, 2.00166582799035419287843852625, 2.00500247912293567011383727501, 2.66745555936155289403605490349, 3.09553092393455181817991695072, 4.14964012363205981019583132946, 4.32451662173143775521012180716, 4.51609018354352792829453870868, 5.32468806028025142523235153501, 5.91877674034631193595653990602, 5.92768647642378177697410623732, 6.39896875523237896851727197586, 6.64516734715479297145623861589, 7.36350503116856883740770103992, 7.38330118863900126867678063803, 8.097128653855452684735595781446, 8.226722828839490055123986141565, 8.949804852530515720052663925626, 9.078603658779419018443433569790, 9.797350943505261902503758501593
