L(s) = 1 | − 2·4-s − 5·9-s + 4·16-s − 25-s − 6·29-s + 10·36-s − 4·37-s + 24·53-s − 8·64-s + 16·81-s + 2·100-s + 14·109-s + 12·113-s + 12·116-s − 13·121-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
L(s) = 1 | − 4-s − 5/3·9-s + 16-s − 1/5·25-s − 1.11·29-s + 5/3·36-s − 0.657·37-s + 3.29·53-s − 64-s + 16/9·81-s + 1/5·100-s + 1.34·109-s + 1.12·113-s + 1.11·116-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−8.072531132972816642931699798914, −7.53353357299583671681693712478, −7.15190569768835151448318214982, −6.56112689904154859098848925311, −5.86914414726978577860225943043, −5.68831226185819751199078977061, −5.32640287451757670345317462029, −4.78851018489545068431737537985, −4.19012039377624132698273056315, −3.64622029282843089012072345623, −3.29982280143545223832366827708, −2.57503205376192440481912334986, −1.99586422105932837771956024974, −0.879757134568912941858458516589, 0,
0.879757134568912941858458516589, 1.99586422105932837771956024974, 2.57503205376192440481912334986, 3.29982280143545223832366827708, 3.64622029282843089012072345623, 4.19012039377624132698273056315, 4.78851018489545068431737537985, 5.32640287451757670345317462029, 5.68831226185819751199078977061, 5.86914414726978577860225943043, 6.56112689904154859098848925311, 7.15190569768835151448318214982, 7.53353357299583671681693712478, 8.072531132972816642931699798914