| L(s) = 1 | − 3-s + 4-s + 2·5-s − 4·7-s − 2·9-s − 12-s − 2·15-s + 16-s + 2·20-s + 4·21-s − 3·25-s + 2·27-s − 4·28-s + 2·29-s + 4·31-s − 8·35-s − 2·36-s − 4·43-s − 4·45-s − 48-s + 2·49-s − 2·60-s − 22·61-s + 8·63-s + 64-s − 20·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 0.894·5-s − 1.51·7-s − 2/3·9-s − 0.288·12-s − 0.516·15-s + 1/4·16-s + 0.447·20-s + 0.872·21-s − 3/5·25-s + 0.384·27-s − 0.755·28-s + 0.371·29-s + 0.718·31-s − 1.35·35-s − 1/3·36-s − 0.609·43-s − 0.596·45-s − 0.144·48-s + 2/7·49-s − 0.258·60-s − 2.81·61-s + 1.00·63-s + 1/8·64-s − 2.37·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 - 14 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−9.165872462453501970889751236873, −8.778478777314452995083184732466, −8.016385905209583936306277548965, −7.61572032202364780377876456843, −6.85559158399622538943520611717, −6.44402481517338937181226066513, −6.10108264916391655699602215932, −5.80480365186012973322542956848, −5.16452074683120544340889617238, −4.49055360066258387336786794879, −3.59148092379748303560705369830, −3.01956314036076486549663593343, −2.49862550273348452232182534380, −1.51170625572432230333101068115, 0,
1.51170625572432230333101068115, 2.49862550273348452232182534380, 3.01956314036076486549663593343, 3.59148092379748303560705369830, 4.49055360066258387336786794879, 5.16452074683120544340889617238, 5.80480365186012973322542956848, 6.10108264916391655699602215932, 6.44402481517338937181226066513, 6.85559158399622538943520611717, 7.61572032202364780377876456843, 8.016385905209583936306277548965, 8.778478777314452995083184732466, 9.165872462453501970889751236873
