| L(s) = 1 | + 2-s + 4-s + 6·7-s + 8-s − 5·9-s + 6·14-s + 16-s + 6·17-s − 5·18-s − 2·23-s + 6·25-s + 6·28-s − 16·31-s + 32-s + 6·34-s − 5·36-s − 16·41-s − 2·46-s + 16·47-s + 13·49-s + 6·50-s + 6·56-s − 16·62-s − 30·63-s + 64-s + 6·68-s + 4·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.26·7-s + 0.353·8-s − 5/3·9-s + 1.60·14-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 0.417·23-s + 6/5·25-s + 1.13·28-s − 2.87·31-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 2.49·41-s − 0.294·46-s + 2.33·47-s + 13/7·49-s + 0.848·50-s + 0.801·56-s − 2.03·62-s − 3.77·63-s + 1/8·64-s + 0.727·68-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372920982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372920982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.44209956582501917351653608378, −9.673084911618933118181680374348, −8.905991841422638352773015657402, −8.408486909011940238981299566497, −8.264845552131759886230905508728, −7.41068992206199799917476516665, −7.25038064772684706053554512263, −6.17646264461499634926610391599, −5.50702009116960948045704530045, −5.28069074518443602737935578596, −4.86338510531971803230901631456, −3.86057015395193111701613139713, −3.27846026315236304587310905813, −2.31224961625843070084602752549, −1.49207601742473373563863301755,
1.49207601742473373563863301755, 2.31224961625843070084602752549, 3.27846026315236304587310905813, 3.86057015395193111701613139713, 4.86338510531971803230901631456, 5.28069074518443602737935578596, 5.50702009116960948045704530045, 6.17646264461499634926610391599, 7.25038064772684706053554512263, 7.41068992206199799917476516665, 8.264845552131759886230905508728, 8.408486909011940238981299566497, 8.905991841422638352773015657402, 9.673084911618933118181680374348, 10.44209956582501917351653608378
