| L(s) = 1 | + 4-s − 6·9-s + 13-s + 16-s + 4·17-s − 8·23-s + 25-s − 4·29-s − 6·36-s − 14·49-s + 52-s + 12·53-s − 4·61-s + 64-s + 4·68-s − 16·79-s + 27·81-s − 8·92-s + 100-s + 12·101-s − 8·103-s − 28·113-s − 4·116-s − 6·117-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2·9-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 1.66·23-s + 1/5·25-s − 0.742·29-s − 36-s − 2·49-s + 0.138·52-s + 1.64·53-s − 0.512·61-s + 1/8·64-s + 0.485·68-s − 1.80·79-s + 3·81-s − 0.834·92-s + 1/10·100-s + 1.19·101-s − 0.788·103-s − 2.63·113-s − 0.371·116-s − 0.554·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.869689837484164888565027122483, −8.172841287289816054273445015073, −7.923578663951581954529383102497, −7.58561732679986769468845152363, −6.69327330977587722422413692301, −6.37597525425947166446541877091, −5.74037388839265127054232811449, −5.56905987734202508880056372110, −4.98983096456527869088395959446, −4.02639003808572429368239555281, −3.56024489190946949262345049451, −2.89815311932047600348194288126, −2.40292181514948168945129765801, −1.47370465322200606140467527719, 0,
1.47370465322200606140467527719, 2.40292181514948168945129765801, 2.89815311932047600348194288126, 3.56024489190946949262345049451, 4.02639003808572429368239555281, 4.98983096456527869088395959446, 5.56905987734202508880056372110, 5.74037388839265127054232811449, 6.37597525425947166446541877091, 6.69327330977587722422413692301, 7.58561732679986769468845152363, 7.923578663951581954529383102497, 8.172841287289816054273445015073, 8.869689837484164888565027122483
