L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 2·13-s + 16-s + 17-s − 4·19-s + 2·20-s − 25-s + 2·26-s − 10·29-s + 8·31-s + 32-s + 34-s − 2·37-s − 4·38-s + 2·40-s + 10·41-s − 12·43-s − 7·49-s − 50-s + 2·52-s − 6·53-s − 10·58-s − 12·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s − 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s − 1.82·43-s − 49-s − 0.141·50-s + 0.277·52-s − 0.824·53-s − 1.31·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.95301400910754, −14.68147434754901, −13.96303019121742, −13.53211524742031, −13.19951122936605, −12.64579543046357, −12.13497104132553, −11.37516824692944, −11.09725293918636, −10.36020287759205, −9.957599803632315, −9.314599391628568, −8.815003584345481, −7.985980518705446, −7.658667351356003, −6.656819144724891, −6.379919321231583, −5.798575803667200, −5.291241457428914, −4.577752653341313, −3.989046612377200, −3.289550490115519, −2.610620958088807, −1.846660678453263, −1.357617127457102, 0,
1.357617127457102, 1.846660678453263, 2.610620958088807, 3.289550490115519, 3.989046612377200, 4.577752653341313, 5.291241457428914, 5.798575803667200, 6.379919321231583, 6.656819144724891, 7.658667351356003, 7.985980518705446, 8.815003584345481, 9.314599391628568, 9.957599803632315, 10.36020287759205, 11.09725293918636, 11.37516824692944, 12.13497104132553, 12.64579543046357, 13.19951122936605, 13.53211524742031, 13.96303019121742, 14.68147434754901, 14.95301400910754