| L(s) = 1 | + 2·2-s + 4-s + 4·11-s − 2·13-s + 16-s − 4·17-s + 8·22-s + 8·23-s − 2·25-s − 4·26-s − 4·29-s − 8·31-s − 2·32-s − 8·34-s − 4·37-s − 16·41-s + 8·43-s + 4·44-s + 16·46-s + 12·47-s − 6·49-s − 4·50-s − 2·52-s + 4·53-s − 8·58-s − 4·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.970·17-s + 1.70·22-s + 1.66·23-s − 2/5·25-s − 0.784·26-s − 0.742·29-s − 1.43·31-s − 0.353·32-s − 1.37·34-s − 0.657·37-s − 2.49·41-s + 1.21·43-s + 0.603·44-s + 2.35·46-s + 1.75·47-s − 6/7·49-s − 0.565·50-s − 0.277·52-s + 0.549·53-s − 1.05·58-s − 0.520·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.950963167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.950963167\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−13.64092450625649755112962206173, −13.44986723938463273555599013326, −12.77900343982831049892648522426, −12.48493999000787113936035322215, −11.93925506234519685751857803358, −11.25744650776750850906437862654, −10.95930482339644359489554990158, −10.27099661688147708313866214628, −9.375624482151927016104097015806, −9.134631821518355283854518142595, −8.549601039417608882680339987542, −7.59361962823542695046904356901, −6.90878693007941002147044931140, −6.66731481470652385178324057492, −5.42948244887922082375027507222, −5.35424162650683405566094675980, −4.37656920736378168473585285560, −3.94635392650984457735444505555, −3.18381189408796018835893862766, −1.87345696301588889971226093835,
1.87345696301588889971226093835, 3.18381189408796018835893862766, 3.94635392650984457735444505555, 4.37656920736378168473585285560, 5.35424162650683405566094675980, 5.42948244887922082375027507222, 6.66731481470652385178324057492, 6.90878693007941002147044931140, 7.59361962823542695046904356901, 8.549601039417608882680339987542, 9.134631821518355283854518142595, 9.375624482151927016104097015806, 10.27099661688147708313866214628, 10.95930482339644359489554990158, 11.25744650776750850906437862654, 11.93925506234519685751857803358, 12.48493999000787113936035322215, 12.77900343982831049892648522426, 13.44986723938463273555599013326, 13.64092450625649755112962206173
