L(s) = 1 | − 3-s + 3·7-s + 9-s + 2·13-s + 17-s − 3·21-s + 23-s − 2·25-s − 2·27-s − 2·39-s + 5·49-s − 51-s + 2·53-s + 3·63-s − 69-s + 2·75-s − 2·79-s + 2·81-s + 3·89-s + 6·91-s − 2·101-s − 2·107-s + 2·117-s + 3·119-s − 121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 3-s + 3·7-s + 9-s + 2·13-s + 17-s − 3·21-s + 23-s − 2·25-s − 2·27-s − 2·39-s + 5·49-s − 51-s + 2·53-s + 3·63-s − 69-s + 2·75-s − 2·79-s + 2·81-s + 3·89-s + 6·91-s − 2·101-s − 2·107-s + 2·117-s + 3·119-s − 121-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.939307088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.939307088\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−8.748119077960381425914505505143, −8.624726037632819273908866235694, −7.965202154151111279028591665667, −7.83808324027748331147129865930, −7.60964343570548200382126728716, −7.28670472666953837477336870987, −6.44941422520343902276979060777, −6.43781592945529871974507883729, −5.63783045366243227147870847505, −5.55859966725343036712703497700, −5.24129221673672671022157121573, −4.96649468017496218892515731289, −4.12561112265413713083337269166, −4.07475688454418425507544440438, −3.87409583793628255342601356442, −3.02693619140794236647783355922, −2.18838843649263246156672064094, −1.80469297793667339808263610921, −1.22069847964624448554316543528, −1.15465668295810960971829390467,
1.15465668295810960971829390467, 1.22069847964624448554316543528, 1.80469297793667339808263610921, 2.18838843649263246156672064094, 3.02693619140794236647783355922, 3.87409583793628255342601356442, 4.07475688454418425507544440438, 4.12561112265413713083337269166, 4.96649468017496218892515731289, 5.24129221673672671022157121573, 5.55859966725343036712703497700, 5.63783045366243227147870847505, 6.43781592945529871974507883729, 6.44941422520343902276979060777, 7.28670472666953837477336870987, 7.60964343570548200382126728716, 7.83808324027748331147129865930, 7.965202154151111279028591665667, 8.624726037632819273908866235694, 8.748119077960381425914505505143