| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s + 4·11-s − 4·14-s − 16-s − 4·22-s + 4·23-s + 2·25-s − 4·28-s + 6·29-s − 5·32-s + 8·37-s − 12·43-s − 4·44-s − 4·46-s + 9·49-s − 2·50-s + 12·56-s − 6·58-s + 7·64-s + 12·67-s + 20·71-s − 8·74-s + 16·77-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s + 1.20·11-s − 1.06·14-s − 1/4·16-s − 0.852·22-s + 0.834·23-s + 2/5·25-s − 0.755·28-s + 1.11·29-s − 0.883·32-s + 1.31·37-s − 1.82·43-s − 0.603·44-s − 0.589·46-s + 9/7·49-s − 0.282·50-s + 1.60·56-s − 0.787·58-s + 7/8·64-s + 1.46·67-s + 2.37·71-s − 0.929·74-s + 1.82·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060409758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060409758\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
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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
−8.001495239315932293788906119901, −7.54704664428304656998737697599, −6.89091370110874460675156333666, −6.65316360102978408419254059580, −6.26519877680992995627667040946, −5.34722580266869021047887825358, −5.15354154727356662727992359462, −4.78497145093938506129125053400, −4.26468288520486434563391484934, −3.88118204664804973206961997329, −3.30230785349223005338544339639, −2.48058162381124778560082497176, −1.86009772382105125183558208073, −1.21500287193424947686578861031, −0.810157386718061727458845988380,
0.810157386718061727458845988380, 1.21500287193424947686578861031, 1.86009772382105125183558208073, 2.48058162381124778560082497176, 3.30230785349223005338544339639, 3.88118204664804973206961997329, 4.26468288520486434563391484934, 4.78497145093938506129125053400, 5.15354154727356662727992359462, 5.34722580266869021047887825358, 6.26519877680992995627667040946, 6.65316360102978408419254059580, 6.89091370110874460675156333666, 7.54704664428304656998737697599, 8.001495239315932293788906119901
