L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s − 2·14-s − 15-s + 16-s + 18-s + 8·19-s + 20-s + 2·21-s − 6·23-s − 24-s + 25-s + 2·26-s − 27-s − 2·28-s + 6·29-s − 30-s + 4·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.205898837\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.205898837\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.34219487919401, −12.13620078724416, −11.77384924287897, −11.19902444029301, −10.75967619879685, −10.19820254110193, −9.853418408011611, −9.602568704841890, −8.862448733576295, −8.363897770789212, −7.808369055113590, −7.296111822766585, −6.748437726280681, −6.366829064449184, −6.040855072389325, −5.449156126318230, −5.109098754063866, −4.593858270212571, −3.919202348242586, −3.448214367411462, −3.016492593759979, −2.444029059396963, −1.630908537994973, −1.223423465086876, −0.4295858447107147,
0.4295858447107147, 1.223423465086876, 1.630908537994973, 2.444029059396963, 3.016492593759979, 3.448214367411462, 3.919202348242586, 4.593858270212571, 5.109098754063866, 5.449156126318230, 6.040855072389325, 6.366829064449184, 6.748437726280681, 7.296111822766585, 7.808369055113590, 8.363897770789212, 8.862448733576295, 9.602568704841890, 9.853418408011611, 10.19820254110193, 10.75967619879685, 11.19902444029301, 11.77384924287897, 12.13620078724416, 12.34219487919401