| L(s) = 1 | − 2·3-s + 2·9-s + 6·11-s + 6·13-s + 8·17-s − 2·19-s − 6·27-s − 6·29-s − 12·33-s + 6·37-s − 12·39-s + 6·43-s − 4·47-s + 14·49-s − 16·51-s + 18·53-s + 4·57-s + 18·59-s − 10·61-s − 6·67-s + 16·79-s + 11·81-s − 18·83-s + 12·87-s − 24·97-s + 12·99-s + 6·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s + 1.80·11-s + 1.66·13-s + 1.94·17-s − 0.458·19-s − 1.15·27-s − 1.11·29-s − 2.08·33-s + 0.986·37-s − 1.92·39-s + 0.914·43-s − 0.583·47-s + 2·49-s − 2.24·51-s + 2.47·53-s + 0.529·57-s + 2.34·59-s − 1.28·61-s − 0.733·67-s + 1.80·79-s + 11/9·81-s − 1.97·83-s + 1.28·87-s − 2.43·97-s + 1.20·99-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237458610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237458610\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−9.702758765679975250072566101202, −9.218389907483313302817842802629, −8.812879179883277481024097487469, −8.542019788782969259470722357142, −7.998065423440694439071413149315, −7.49209862717289665177562414446, −7.13308311053660077800456212812, −6.78121466765948119591514506054, −6.14098217292904939195342161985, −5.97919179546714795891094934094, −5.51278277207434065107806723603, −5.49392581475311240114600842940, −4.51333780310260280007689941975, −3.98189993015752885571722268460, −3.85953434011924264483549117304, −3.44620474159706869408090868779, −2.56127311830190730740249320736, −1.76296657973499314889894800984, −1.13508494439198558991427077677, −0.78693737983005232923352362527,
0.78693737983005232923352362527, 1.13508494439198558991427077677, 1.76296657973499314889894800984, 2.56127311830190730740249320736, 3.44620474159706869408090868779, 3.85953434011924264483549117304, 3.98189993015752885571722268460, 4.51333780310260280007689941975, 5.49392581475311240114600842940, 5.51278277207434065107806723603, 5.97919179546714795891094934094, 6.14098217292904939195342161985, 6.78121466765948119591514506054, 7.13308311053660077800456212812, 7.49209862717289665177562414446, 7.998065423440694439071413149315, 8.542019788782969259470722357142, 8.812879179883277481024097487469, 9.218389907483313302817842802629, 9.702758765679975250072566101202
