| L(s) = 1 | − 4-s + 3·7-s − 2·9-s + 5·13-s + 16-s + 2·23-s − 4·25-s − 3·28-s + 3·29-s + 2·36-s + 5·49-s − 5·52-s + 12·53-s + 6·59-s − 6·63-s − 64-s − 19·67-s − 2·71-s − 5·81-s − 7·83-s + 15·91-s − 2·92-s + 4·100-s − 13·103-s + 17·107-s − 16·109-s + 3·112-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.13·7-s − 2/3·9-s + 1.38·13-s + 1/4·16-s + 0.417·23-s − 4/5·25-s − 0.566·28-s + 0.557·29-s + 1/3·36-s + 5/7·49-s − 0.693·52-s + 1.64·53-s + 0.781·59-s − 0.755·63-s − 1/8·64-s − 2.32·67-s − 0.237·71-s − 5/9·81-s − 0.768·83-s + 1.57·91-s − 0.208·92-s + 2/5·100-s − 1.28·103-s + 1.64·107-s − 1.53·109-s + 0.283·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113220165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113220165\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
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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
−11.10095789490431463214003897592, −10.49128616852238001284277984628, −10.09750511866670223720955260059, −9.167263130908966263557714337576, −8.779448746338751367140700661297, −8.370064452758151162057447005505, −7.84633911458726698508718324403, −7.18123333603327178916566455468, −6.31083125169409972137627220056, −5.70994626755523664832649392309, −5.18460319739737837111699606789, −4.34582199427559466743138441203, −3.75640671227168289341490186247, −2.70027648215823123322990139492, −1.38957076416327272839190174600,
1.38957076416327272839190174600, 2.70027648215823123322990139492, 3.75640671227168289341490186247, 4.34582199427559466743138441203, 5.18460319739737837111699606789, 5.70994626755523664832649392309, 6.31083125169409972137627220056, 7.18123333603327178916566455468, 7.84633911458726698508718324403, 8.370064452758151162057447005505, 8.779448746338751367140700661297, 9.167263130908966263557714337576, 10.09750511866670223720955260059, 10.49128616852238001284277984628, 11.10095789490431463214003897592
