| L(s) = 1 | + 3·2-s + 5·4-s − 6·5-s + 14·7-s + 27·8-s − 18·10-s + 6·11-s + 16·13-s + 42·14-s + 69·16-s + 6·17-s + 64·19-s − 30·20-s + 18·22-s − 6·23-s − 166·25-s + 48·26-s + 70·28-s + 252·29-s + 40·31-s + 27·32-s + 18·34-s − 84·35-s − 248·37-s + 192·38-s − 162·40-s + 450·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 5/8·4-s − 0.536·5-s + 0.755·7-s + 1.19·8-s − 0.569·10-s + 0.164·11-s + 0.341·13-s + 0.801·14-s + 1.07·16-s + 0.0856·17-s + 0.772·19-s − 0.335·20-s + 0.174·22-s − 0.0543·23-s − 1.32·25-s + 0.362·26-s + 0.472·28-s + 1.61·29-s + 0.231·31-s + 0.149·32-s + 0.0907·34-s − 0.405·35-s − 1.10·37-s + 0.819·38-s − 0.640·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.442532633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.442532633\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{2} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 6 T + 202 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 1246 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 16 T + 2406 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 9778 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 7870 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 450 T + 175642 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 141790 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1104 T + 602230 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 804 T + 380614 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 954 T + 13106 p T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1944 T + 1957030 T^{2} + 1944 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 366 T + 1156090 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−14.48289404713994943445822704582, −14.13601139323524478233999075152, −13.70142794982341405926684776989, −13.32097880730821409169839349298, −12.38185296883303587616969433143, −12.07154865082848865245391045155, −11.54804003470768597486730632527, −10.83622543175646532093228205860, −10.49999815706740550922325046677, −9.628205526546974018915353797586, −8.785386662532361036313842293831, −8.023929318689198866989727594526, −7.54391036600959731955986959855, −6.91596543508203836799506883912, −5.85166208683984172394022522251, −5.29172520676447856530071674478, −4.26703506910866712419780518048, −4.06663314762340927113853137964, −2.72503130760728466642393564230, −1.33082132098684921023904049139,
1.33082132098684921023904049139, 2.72503130760728466642393564230, 4.06663314762340927113853137964, 4.26703506910866712419780518048, 5.29172520676447856530071674478, 5.85166208683984172394022522251, 6.91596543508203836799506883912, 7.54391036600959731955986959855, 8.023929318689198866989727594526, 8.785386662532361036313842293831, 9.628205526546974018915353797586, 10.49999815706740550922325046677, 10.83622543175646532093228205860, 11.54804003470768597486730632527, 12.07154865082848865245391045155, 12.38185296883303587616969433143, 13.32097880730821409169839349298, 13.70142794982341405926684776989, 14.13601139323524478233999075152, 14.48289404713994943445822704582
