L(s) = 1 | + 2-s − 3-s + 4-s + 1.42·5-s − 6-s − 1.54·7-s + 8-s + 9-s + 1.42·10-s − 1.53·11-s − 12-s + 2.48·13-s − 1.54·14-s − 1.42·15-s + 16-s − 17-s + 18-s + 4.44·19-s + 1.42·20-s + 1.54·21-s − 1.53·22-s − 4.57·23-s − 24-s − 2.95·25-s + 2.48·26-s − 27-s − 1.54·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.639·5-s − 0.408·6-s − 0.582·7-s + 0.353·8-s + 0.333·9-s + 0.451·10-s − 0.461·11-s − 0.288·12-s + 0.688·13-s − 0.411·14-s − 0.369·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.01·19-s + 0.319·20-s + 0.336·21-s − 0.326·22-s − 0.953·23-s − 0.204·24-s − 0.591·25-s + 0.486·26-s − 0.192·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 + 8.05T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 0.0454T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.45347347591554903996870083706, −6.89165887430224373026460666990, −6.04331799263065405730520420126, −5.60538960770118313405882173423, −5.11136673252942850588138459034, −3.89379341890378324204284993152, −3.49966439335480038717597918224, −2.29766561128742883722440417668, −1.53440484652456537290710192189, 0,
1.53440484652456537290710192189, 2.29766561128742883722440417668, 3.49966439335480038717597918224, 3.89379341890378324204284993152, 5.11136673252942850588138459034, 5.60538960770118313405882173423, 6.04331799263065405730520420126, 6.89165887430224373026460666990, 7.45347347591554903996870083706