| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s − 6·12-s + 2·13-s + 5·16-s + 4·17-s − 6·18-s + 8·24-s − 4·26-s − 4·27-s − 12·29-s + 8·31-s − 6·32-s − 8·34-s + 9·36-s + 12·37-s − 4·39-s − 4·41-s − 8·43-s + 16·47-s − 10·48-s − 6·49-s − 8·51-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s − 1.73·12-s + 0.554·13-s + 5/4·16-s + 0.970·17-s − 1.41·18-s + 1.63·24-s − 0.784·26-s − 0.769·27-s − 2.22·29-s + 1.43·31-s − 1.06·32-s − 1.37·34-s + 3/2·36-s + 1.97·37-s − 0.640·39-s − 0.624·41-s − 1.21·43-s + 2.33·47-s − 1.44·48-s − 6/7·49-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7456958880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7456958880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.381119483580321670414397939315, −9.219115903038074626288763489422, −8.497802958648184043353661337669, −8.243409452173305058300395212591, −7.75916640616896204146584076146, −7.61867841167693829596398597289, −6.88180835921894957660755905301, −6.88064381881974669853439768497, −6.18436617040436159419623089311, −5.92468674655933683902335830682, −5.56731529714752289199474351991, −5.19307669183466344828246987613, −4.36491764852097863929258806300, −4.21175510665677260094557632268, −3.27662068984700935406229227369, −3.13046651485550034973767298423, −2.15400126718019764932313057866, −1.73887831165524401257091125502, −1.01601587301387497534768664382, −0.52886766222493632297846208002,
0.52886766222493632297846208002, 1.01601587301387497534768664382, 1.73887831165524401257091125502, 2.15400126718019764932313057866, 3.13046651485550034973767298423, 3.27662068984700935406229227369, 4.21175510665677260094557632268, 4.36491764852097863929258806300, 5.19307669183466344828246987613, 5.56731529714752289199474351991, 5.92468674655933683902335830682, 6.18436617040436159419623089311, 6.88064381881974669853439768497, 6.88180835921894957660755905301, 7.61867841167693829596398597289, 7.75916640616896204146584076146, 8.243409452173305058300395212591, 8.497802958648184043353661337669, 9.219115903038074626288763489422, 9.381119483580321670414397939315
