| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 9-s + 10-s + 13-s + 16-s − 17-s − 18-s − 20-s − 4·25-s − 26-s − 4·29-s − 32-s + 34-s + 36-s + 3·37-s + 40-s − 41-s − 45-s + 2·49-s + 4·50-s + 52-s + 4·53-s + 4·58-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.277·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.223·20-s − 4/5·25-s − 0.196·26-s − 0.742·29-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.493·37-s + 0.158·40-s − 0.156·41-s − 0.149·45-s + 2/7·49-s + 0.565·50-s + 0.138·52-s + 0.549·53-s + 0.525·58-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−7.71022522256589416564546131041, −7.53559951048437104753723960691, −7.06701276440777301064394019700, −6.52971342433885619293464232505, −6.17979078294253061454083616078, −5.64427442990375843143537531479, −5.24059866116821516787115840226, −4.44350916295648781392940506816, −4.22310356317854642220436567177, −3.52255094473294927023542270272, −3.12642006657231163394562618465, −2.31546529967749138889055589824, −1.81472585419752838581456710383, −1.01285298341888206124189479357, 0,
1.01285298341888206124189479357, 1.81472585419752838581456710383, 2.31546529967749138889055589824, 3.12642006657231163394562618465, 3.52255094473294927023542270272, 4.22310356317854642220436567177, 4.44350916295648781392940506816, 5.24059866116821516787115840226, 5.64427442990375843143537531479, 6.17979078294253061454083616078, 6.52971342433885619293464232505, 7.06701276440777301064394019700, 7.53559951048437104753723960691, 7.71022522256589416564546131041
