| L(s) = 1 | + 12·3-s + 90·9-s − 72·13-s + 216·17-s − 120·23-s + 228·25-s + 540·27-s − 48·29-s − 864·39-s − 1.06e3·43-s + 328·49-s + 2.59e3·51-s − 864·53-s − 2.28e3·61-s − 1.44e3·69-s + 2.73e3·75-s − 288·79-s + 2.83e3·81-s − 576·87-s + 528·101-s + 3.24e3·107-s + 1.41e3·113-s − 6.48e3·117-s + 4.89e3·121-s + 127-s − 1.27e4·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s − 1.53·13-s + 3.08·17-s − 1.08·23-s + 1.82·25-s + 3.84·27-s − 0.307·29-s − 3.54·39-s − 3.77·43-s + 0.956·49-s + 7.11·51-s − 2.23·53-s − 4.78·61-s − 2.51·69-s + 4.21·75-s − 0.410·79-s + 35/9·81-s − 0.709·87-s + 0.520·101-s + 2.92·107-s + 1.17·113-s − 5.12·117-s + 3.67·121-s + 0.000698·127-s − 8.71·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.26317411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.26317411\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 238 p T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 228 T^{2} + 33862 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 328 T^{2} + 51918 T^{4} - 328 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 4896 T^{2} + 9533230 T^{4} - 4896 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 15880 T^{2} + 155242878 T^{4} - 15880 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 60 T + 1870 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 25558 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 102136 T^{2} + 4357504590 T^{4} - 102136 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 2164 T^{2} - 2618990922 T^{4} - 2164 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 230292 T^{2} + 22372016182 T^{4} - 230292 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 532 T + 206406 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 85056 T^{2} + 23167872958 T^{4} + 85056 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 432 T + 134134 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 716592 T^{2} + 212420895502 T^{4} - 716592 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 1140 T + 12598 p T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 475384 T^{2} + 105182398878 T^{4} - 475384 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1283952 T^{2} + 666179277502 T^{4} - 1283952 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 762916 T^{2} + 359882924838 T^{4} - 762916 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 144 T + 825118 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 1633872 T^{2} + 191792062 p^{2} T^{4} - 1633872 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 2759412 T^{2} + 2896886558902 T^{4} - 2759412 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 53564 T^{2} + 1619815156998 T^{4} + 53564 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41928739117753195811183988116, −7.19711239364937855471408939247, −6.97354161412870613435908502621, −6.40320307284319038947596454393, −6.36847666077797663998722632223, −6.08226248877107951965765245845, −5.95209668524552188979377490409, −5.27044321358726813992726444940, −5.20530948502973825053167109051, −4.92978838377272984217559755949, −4.76642137098643650438497824485, −4.51280722826644570525197705958, −4.15527704850992166861336097085, −3.82120693684665394280740135619, −3.34532053258555974829514076124, −3.26544945726875844312317123818, −3.14086208813471353916711029777, −2.95146036054364565719049211509, −2.67954778648324418870897172717, −1.95842475088113614390088014322, −1.90980466673194390115971276989, −1.58699838158884966318774668602, −1.34606562017540056003782783683, −0.70266743699976450179021056113, −0.36395221648109986049947158094,
0.36395221648109986049947158094, 0.70266743699976450179021056113, 1.34606562017540056003782783683, 1.58699838158884966318774668602, 1.90980466673194390115971276989, 1.95842475088113614390088014322, 2.67954778648324418870897172717, 2.95146036054364565719049211509, 3.14086208813471353916711029777, 3.26544945726875844312317123818, 3.34532053258555974829514076124, 3.82120693684665394280740135619, 4.15527704850992166861336097085, 4.51280722826644570525197705958, 4.76642137098643650438497824485, 4.92978838377272984217559755949, 5.20530948502973825053167109051, 5.27044321358726813992726444940, 5.95209668524552188979377490409, 6.08226248877107951965765245845, 6.36847666077797663998722632223, 6.40320307284319038947596454393, 6.97354161412870613435908502621, 7.19711239364937855471408939247, 7.41928739117753195811183988116
