| L(s) = 1 | + 2·2-s + 3·4-s − 3·7-s + 4·8-s − 9-s + 2·11-s − 6·14-s + 5·16-s − 2·18-s + 4·22-s − 7·23-s + 4·25-s − 9·28-s − 6·29-s + 6·32-s − 3·36-s − 12·37-s + 11·43-s + 6·44-s − 14·46-s + 2·49-s + 8·50-s − 2·53-s − 12·56-s − 12·58-s + 3·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.13·7-s + 1.41·8-s − 1/3·9-s + 0.603·11-s − 1.60·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s − 1.45·23-s + 4/5·25-s − 1.70·28-s − 1.11·29-s + 1.06·32-s − 1/2·36-s − 1.97·37-s + 1.67·43-s + 0.904·44-s − 2.06·46-s + 2/7·49-s + 1.13·50-s − 0.274·53-s − 1.60·56-s − 1.57·58-s + 0.377·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088129922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088129922\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
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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
−11.46608213493968626727300163207, −10.82760088508319938062447305558, −10.33657435048486602488996587889, −9.695683921864892940047753506823, −9.112125576411928642769877559337, −8.439644640945821413931531157457, −7.57452056108471410921263826525, −7.01219029556816612797314956669, −6.41820765993181121572351111656, −5.90511758387739515136791413173, −5.34383943955583570994254221153, −4.40308216215535518235923432343, −3.70870276908503450949320302427, −3.15624900586311890679207826246, −2.06411574815739123744459976512,
2.06411574815739123744459976512, 3.15624900586311890679207826246, 3.70870276908503450949320302427, 4.40308216215535518235923432343, 5.34383943955583570994254221153, 5.90511758387739515136791413173, 6.41820765993181121572351111656, 7.01219029556816612797314956669, 7.57452056108471410921263826525, 8.439644640945821413931531157457, 9.112125576411928642769877559337, 9.695683921864892940047753506823, 10.33657435048486602488996587889, 10.82760088508319938062447305558, 11.46608213493968626727300163207
