| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 4·7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 6·13-s + 4·14-s − 15-s + 16-s + 17-s + 18-s − 4·19-s − 20-s + 4·21-s − 22-s + 4·23-s + 24-s + 25-s + 6·26-s + 27-s + 4·28-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 − 0.301·11-s + 0.288·12-s + 1.66·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.213·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.907306266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.907306266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.947570864653798640779720627410, −7.69997267745214092169320208816, −6.66279296607404645133921525925, −5.92914159185507930994870682464, −5.07740646319618522089224833750, −4.39619345279440563260868038705, −3.80223924063613251560490157562, −2.94703026097533759358311423056, −1.93279827462888311511080732071, −1.14832964461223094305421892197,
1.14832964461223094305421892197, 1.93279827462888311511080732071, 2.94703026097533759358311423056, 3.80223924063613251560490157562, 4.39619345279440563260868038705, 5.07740646319618522089224833750, 5.92914159185507930994870682464, 6.66279296607404645133921525925, 7.69997267745214092169320208816, 7.947570864653798640779720627410
