L(s) = 1 | + 7-s + 9-s + 2·11-s − 6·23-s + 2·25-s − 4·29-s + 16·37-s − 4·43-s + 49-s + 4·53-s + 63-s − 8·67-s + 2·71-s + 2·77-s + 8·79-s + 81-s + 2·99-s + 6·107-s + 8·109-s + 20·113-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.25·23-s + 2/5·25-s − 0.742·29-s + 2.63·37-s − 0.609·43-s + 1/7·49-s + 0.549·53-s + 0.125·63-s − 0.977·67-s + 0.237·71-s + 0.227·77-s + 0.900·79-s + 1/9·81-s + 0.201·99-s + 0.580·107-s + 0.766·109-s + 1.88·113-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.169153816\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.169153816\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
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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
−8.240739862017027813665018066392, −7.72915489024576600570511211044, −7.50703712292892772359120089300, −6.94148655169425245403736578361, −6.36253111156959396996583277514, −6.10167714402402822547799176650, −5.58968845531103875583597421724, −5.01166982010850614499698079883, −4.42515283972218750802531686304, −4.13390765838015640821056725358, −3.57667722576508697923406002714, −2.88811232495259139240955314475, −2.19650743855414635123430423580, −1.62004487044978892864679486425, −0.73842679818366932608286887376,
0.73842679818366932608286887376, 1.62004487044978892864679486425, 2.19650743855414635123430423580, 2.88811232495259139240955314475, 3.57667722576508697923406002714, 4.13390765838015640821056725358, 4.42515283972218750802531686304, 5.01166982010850614499698079883, 5.58968845531103875583597421724, 6.10167714402402822547799176650, 6.36253111156959396996583277514, 6.94148655169425245403736578361, 7.50703712292892772359120089300, 7.72915489024576600570511211044, 8.240739862017027813665018066392