| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 2·11-s + 4·13-s − 2·14-s + 16-s − 6·17-s − 4·19-s − 20-s − 2·22-s + 23-s + 25-s + 4·26-s − 2·28-s − 2·29-s − 4·31-s + 32-s − 6·34-s + 2·35-s − 4·37-s − 4·38-s − 40-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.338·35-s − 0.657·37-s − 0.648·38-s − 0.158·40-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.649711311063082814366457165938, −7.961669236426023277906867131666, −6.84816031514451450064663573243, −6.46122606544515908105196529361, −5.52458016649347988292471411852, −4.55628418651765788578211312788, −3.78383307097973358869486845562, −2.98444748195887011167713493304, −1.85164441327250887995307415846, 0,
1.85164441327250887995307415846, 2.98444748195887011167713493304, 3.78383307097973358869486845562, 4.55628418651765788578211312788, 5.52458016649347988292471411852, 6.46122606544515908105196529361, 6.84816031514451450064663573243, 7.961669236426023277906867131666, 8.649711311063082814366457165938
