| L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·15-s − 12·19-s − 25-s − 27-s − 4·43-s − 2·45-s + 8·47-s + 10·49-s − 4·53-s + 12·57-s + 4·67-s − 12·71-s − 8·73-s + 75-s + 81-s + 24·95-s + 12·97-s + 12·101-s − 10·121-s + 12·125-s + 127-s + 4·129-s + 131-s + 2·135-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s − 2.75·19-s − 1/5·25-s − 0.192·27-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 10/7·49-s − 0.549·53-s + 1.58·57-s + 0.488·67-s − 1.42·71-s − 0.936·73-s + 0.115·75-s + 1/9·81-s + 2.46·95-s + 1.21·97-s + 1.19·101-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.172·135-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6831964253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6831964253\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.796523279169400324099811274199, −8.311117487892413678982025088035, −7.71916516278321719237455267489, −7.45291882652318289749379821721, −6.74822994356548814377277210941, −6.50122389996814390419899237868, −5.88978115346659328854413481568, −5.51271162729538474767227340615, −4.67629508369109499467730348616, −4.30042216411982689284177737617, −3.99884352876894801076363929522, −3.25909981327213640380196307522, −2.40621497002503574889419611187, −1.77015580864553681627788006708, −0.47534095433036505976960674830,
0.47534095433036505976960674830, 1.77015580864553681627788006708, 2.40621497002503574889419611187, 3.25909981327213640380196307522, 3.99884352876894801076363929522, 4.30042216411982689284177737617, 4.67629508369109499467730348616, 5.51271162729538474767227340615, 5.88978115346659328854413481568, 6.50122389996814390419899237868, 6.74822994356548814377277210941, 7.45291882652318289749379821721, 7.71916516278321719237455267489, 8.311117487892413678982025088035, 8.796523279169400324099811274199
