L(s) = 1 | + 2-s − 3-s + 4-s − 0.230·5-s − 6-s + 8-s + 9-s − 0.230·10-s − 2.27·11-s − 12-s − 4.59·13-s + 0.230·15-s + 16-s − 17-s + 18-s + 1.67·19-s − 0.230·20-s − 2.27·22-s + 3.08·23-s − 24-s − 4.94·25-s − 4.59·26-s − 27-s + 0.845·29-s + 0.230·30-s + 0.325·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.102·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.0728·10-s − 0.685·11-s − 0.288·12-s − 1.27·13-s + 0.0594·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.384·19-s − 0.0514·20-s − 0.484·22-s + 0.644·23-s − 0.204·24-s − 0.989·25-s − 0.901·26-s − 0.192·27-s + 0.157·29-s + 0.0420·30-s + 0.0584·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.050260396\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.050260396\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 0.230T + 5T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 0.845T + 29T^{2} \) |
| 31 | \( 1 - 0.325T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 0.0829T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 + 0.733T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.20T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 4.03T + 97T^{2} \) |
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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
−7.82000023115817726651642153099, −7.53902021757904441353291427338, −6.71815584198075163615518558123, −5.91497966297248691554036162452, −5.28295875625396308104947920436, −4.65739455547330888797301532278, −3.93005344594336094239255524606, −2.81946188605194353411324035125, −2.15707228342791368415251502666, −0.70337227888120997384075875240,
0.70337227888120997384075875240, 2.15707228342791368415251502666, 2.81946188605194353411324035125, 3.93005344594336094239255524606, 4.65739455547330888797301532278, 5.28295875625396308104947920436, 5.91497966297248691554036162452, 6.71815584198075163615518558123, 7.53902021757904441353291427338, 7.82000023115817726651642153099