| L(s) = 1 | − 2·3-s + 4-s + 9-s − 6·11-s − 2·12-s + 16-s − 12·23-s + 4·27-s + 4·31-s + 12·33-s + 36-s + 10·37-s − 6·44-s − 2·48-s − 10·49-s + 6·53-s − 6·59-s + 64-s + 4·67-s + 24·69-s − 12·71-s − 11·81-s + 30·89-s − 12·92-s − 8·93-s + 10·97-s − 6·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 1/4·16-s − 2.50·23-s + 0.769·27-s + 0.718·31-s + 2.08·33-s + 1/6·36-s + 1.64·37-s − 0.904·44-s − 0.288·48-s − 1.42·49-s + 0.824·53-s − 0.781·59-s + 1/8·64-s + 0.488·67-s + 2.88·69-s − 1.42·71-s − 1.22·81-s + 3.17·89-s − 1.25·92-s − 0.829·93-s + 1.01·97-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.51418040596748774281495271118, −6.87270890517953419594923571997, −6.33912960567205846581832395049, −6.16589221432355354770528662392, −5.72823291810368110448034634176, −5.45392561750926158713223148879, −4.79009769782035437981683210932, −4.60961319984996389968509073361, −3.99345182068769795571103477737, −3.31795145100301636360920191493, −2.76895688441925525066222597758, −2.31291952514841725707466337624, −1.76058339222977515115768094302, −0.74426691236933287668136511202, 0,
0.74426691236933287668136511202, 1.76058339222977515115768094302, 2.31291952514841725707466337624, 2.76895688441925525066222597758, 3.31795145100301636360920191493, 3.99345182068769795571103477737, 4.60961319984996389968509073361, 4.79009769782035437981683210932, 5.45392561750926158713223148879, 5.72823291810368110448034634176, 6.16589221432355354770528662392, 6.33912960567205846581832395049, 6.87270890517953419594923571997, 7.51418040596748774281495271118
