L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 6·11-s + 2·12-s − 2·13-s + 16-s + 17-s + 18-s + 4·19-s + 6·22-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·31-s + 32-s + 12·33-s + 34-s + 36-s − 4·37-s + 4·38-s − 4·39-s − 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 1.27·22-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.718·31-s + 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s − 0.657·37-s + 0.648·38-s − 0.640·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.227551659\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.227551659\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.435449454754526833910951528272, −8.550482821819457584609282080376, −7.75961744967946030738916282437, −6.98368217520104403156128027372, −6.16055468076817207109148001002, −5.16488039277883802975012518677, −4.04100934533934329070442787144, −3.50747685387131529534531471772, −2.52652883309992049334898398402, −1.44781063014612234378860428910,
1.44781063014612234378860428910, 2.52652883309992049334898398402, 3.50747685387131529534531471772, 4.04100934533934329070442787144, 5.16488039277883802975012518677, 6.16055468076817207109148001002, 6.98368217520104403156128027372, 7.75961744967946030738916282437, 8.550482821819457584609282080376, 9.435449454754526833910951528272