| L(s) = 1 | − 2·2-s + 2·4-s − 4·7-s − 9-s + 8·14-s − 4·16-s + 12·17-s + 2·18-s + 8·23-s − 8·28-s + 20·31-s + 8·32-s − 24·34-s − 2·36-s + 20·41-s − 16·46-s + 8·47-s − 2·49-s − 40·62-s + 4·63-s − 8·64-s + 24·68-s − 8·71-s − 20·73-s − 28·79-s + 81-s − 40·82-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s + 2.13·14-s − 16-s + 2.91·17-s + 0.471·18-s + 1.66·23-s − 1.51·28-s + 3.59·31-s + 1.41·32-s − 4.11·34-s − 1/3·36-s + 3.12·41-s − 2.35·46-s + 1.16·47-s − 2/7·49-s − 5.08·62-s + 0.503·63-s − 64-s + 2.91·68-s − 0.949·71-s − 2.34·73-s − 3.15·79-s + 1/9·81-s − 4.41·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8627505691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8627505691\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.46982852197989060389221127174, −10.34519256248327191045397691680, −9.861859387777574106379461685309, −9.760757295120013946296302677340, −9.026677707446116323796825846477, −8.970076493502703952046791255288, −8.263917831551586856894272402927, −7.83727342278446855732740151019, −7.41860431444945416935591347114, −7.15133876144596789457034576020, −6.36429654460564020163414918027, −6.05334046248431055581744507227, −5.69250384819205677507488475638, −4.78342985241194947751533208569, −4.38171239632414891983878406977, −3.41369744806173821743501061303, −2.91095070327648527113839263551, −2.66475744322071252239201683237, −1.15080276500926866166428846606, −0.824933504831208538900601064344,
0.824933504831208538900601064344, 1.15080276500926866166428846606, 2.66475744322071252239201683237, 2.91095070327648527113839263551, 3.41369744806173821743501061303, 4.38171239632414891983878406977, 4.78342985241194947751533208569, 5.69250384819205677507488475638, 6.05334046248431055581744507227, 6.36429654460564020163414918027, 7.15133876144596789457034576020, 7.41860431444945416935591347114, 7.83727342278446855732740151019, 8.263917831551586856894272402927, 8.970076493502703952046791255288, 9.026677707446116323796825846477, 9.760757295120013946296302677340, 9.861859387777574106379461685309, 10.34519256248327191045397691680, 10.46982852197989060389221127174
