| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 3·11-s − 2·12-s − 14-s + 16-s − 3·17-s + 18-s − 6·19-s + 2·21-s + 3·22-s − 2·23-s − 2·24-s + 4·27-s − 28-s − 3·29-s + 3·31-s + 32-s − 6·33-s − 3·34-s + 36-s + 37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.37·19-s + 0.436·21-s + 0.639·22-s − 0.417·23-s − 0.408·24-s + 0.769·27-s − 0.188·28-s − 0.557·29-s + 0.538·31-s + 0.176·32-s − 1.04·33-s − 0.514·34-s + 1/6·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.909285651436971781795443789502, −7.940617253362879027486346170110, −6.74330103812297173688363684265, −6.38244555304273679459795010368, −5.75894180480402141133249940840, −4.68567966081196923689329880264, −4.14074597563837066006181115129, −2.94631810557103912586946477936, −1.64514204035112192009306844052, 0,
1.64514204035112192009306844052, 2.94631810557103912586946477936, 4.14074597563837066006181115129, 4.68567966081196923689329880264, 5.75894180480402141133249940840, 6.38244555304273679459795010368, 6.74330103812297173688363684265, 7.940617253362879027486346170110, 8.909285651436971781795443789502
