| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 4·7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s + 2·13-s − 4·14-s − 15-s + 16-s − 2·17-s − 18-s − 20-s + 4·21-s − 4·22-s − 24-s + 25-s − 2·26-s + 27-s + 4·28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719950421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719950421\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−19.17862991268810, −18.67528047226653, −17.88753799106819, −17.30847761252781, −16.66603845631470, −15.69820375628860, −15.11878886802958, −14.44081018365578, −13.91781819941612, −12.83791122781310, −11.84197248268531, −11.33995646763354, −10.73127059514354, −9.643282737274141, −8.765464851089721, −8.429871555930801, −7.522432900575890, −6.854430933746538, −5.646900947622272, −4.424553917381870, −3.652665872522307, −2.156653661573234, −1.214359045040775,
1.214359045040775, 2.156653661573234, 3.652665872522307, 4.424553917381870, 5.646900947622272, 6.854430933746538, 7.522432900575890, 8.429871555930801, 8.765464851089721, 9.643282737274141, 10.73127059514354, 11.33995646763354, 11.84197248268531, 12.83791122781310, 13.91781819941612, 14.44081018365578, 15.11878886802958, 15.69820375628860, 16.66603845631470, 17.30847761252781, 17.88753799106819, 18.67528047226653, 19.17862991268810
