| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s + 4·11-s + 4·13-s + 8·17-s − 4·21-s + 4·23-s + 2·25-s + 4·27-s + 4·29-s − 8·31-s + 8·33-s − 4·37-s + 8·39-s + 16·41-s − 16·43-s + 8·47-s + 3·49-s + 16·51-s − 4·53-s − 16·59-s − 4·61-s − 6·63-s + 8·67-s + 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s + 1.20·11-s + 1.10·13-s + 1.94·17-s − 0.872·21-s + 0.834·23-s + 2/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s + 1.39·33-s − 0.657·37-s + 1.28·39-s + 2.49·41-s − 2.43·43-s + 1.16·47-s + 3/7·49-s + 2.24·51-s − 0.549·53-s − 2.08·59-s − 0.512·61-s − 0.755·63-s + 0.977·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.511791915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.511791915\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.54037996881590863937881336647, −10.33766677607814470408512191775, −9.514613460313556899168862854738, −9.379615572406319797328090655477, −9.225638526702577970973858034414, −8.563282874025115237795605104710, −8.109239953343344007279947434768, −7.87523126180874102016645820385, −7.00469529776766762628040782257, −7.00413869600387648617253226188, −6.33507440054543428246941670987, −5.84833168662077872258392685058, −5.33227847595585265381325435022, −4.65005603433511873763283667591, −3.79438648256174728478462858442, −3.73204424624066476989619424617, −3.13650170611606548034931844134, −2.65386364635551658183695782130, −1.53667169342930680116766233112, −1.11697054605639928647811921199,
1.11697054605639928647811921199, 1.53667169342930680116766233112, 2.65386364635551658183695782130, 3.13650170611606548034931844134, 3.73204424624066476989619424617, 3.79438648256174728478462858442, 4.65005603433511873763283667591, 5.33227847595585265381325435022, 5.84833168662077872258392685058, 6.33507440054543428246941670987, 7.00413869600387648617253226188, 7.00469529776766762628040782257, 7.87523126180874102016645820385, 8.109239953343344007279947434768, 8.563282874025115237795605104710, 9.225638526702577970973858034414, 9.379615572406319797328090655477, 9.514613460313556899168862854738, 10.33766677607814470408512191775, 10.54037996881590863937881336647
