| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 12-s + 13-s − 2·14-s + 15-s + 16-s + 4·17-s + 18-s − 19-s − 20-s + 2·21-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 29-s + 30-s + 8·31-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 + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.185·29-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832402192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832402192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.67313787905140, −13.95761178015670, −13.52126373239129, −13.04326542357133, −12.42389759241857, −12.05030611101257, −11.76802218086100, −10.91477764152413, −10.64928050240380, −9.984439076985129, −9.539917662201771, −8.777591579567936, −8.147898872080499, −7.505138858180953, −7.089988224895415, −6.347344057057339, −5.988963111112104, −5.488274304415824, −4.671227519814652, −4.229678939009715, −3.606476450798318, −2.955524725703409, −2.371264180782139, −1.231053973248549, −0.6096456381877584,
0.6096456381877584, 1.231053973248549, 2.371264180782139, 2.955524725703409, 3.606476450798318, 4.229678939009715, 4.671227519814652, 5.488274304415824, 5.988963111112104, 6.347344057057339, 7.089988224895415, 7.505138858180953, 8.147898872080499, 8.777591579567936, 9.539917662201771, 9.984439076985129, 10.64928050240380, 10.91477764152413, 11.76802218086100, 12.05030611101257, 12.42389759241857, 13.04326542357133, 13.52126373239129, 13.95761178015670, 14.67313787905140
