| L(s) = 1 | − 2-s + 0.486·3-s + 4-s − 0.616·5-s − 0.486·6-s − 8-s − 2.76·9-s + 0.616·10-s − 1.68·11-s + 0.486·12-s − 5.11·13-s − 0.299·15-s + 16-s + 4.37·17-s + 2.76·18-s + 7.65·19-s − 0.616·20-s + 1.68·22-s + 2.40·23-s − 0.486·24-s − 4.62·25-s + 5.11·26-s − 2.80·27-s + 4.00·29-s + 0.299·30-s − 4.02·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.280·3-s + 0.5·4-s − 0.275·5-s − 0.198·6-s − 0.353·8-s − 0.921·9-s + 0.194·10-s − 0.506·11-s + 0.140·12-s − 1.41·13-s − 0.0773·15-s + 0.250·16-s + 1.06·17-s + 0.651·18-s + 1.75·19-s − 0.137·20-s + 0.358·22-s + 0.501·23-s − 0.0992·24-s − 0.924·25-s + 1.00·26-s − 0.539·27-s + 0.744·29-s + 0.0547·30-s − 0.722·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9850018984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9850018984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.486T + 3T^{2} \) |
| 5 | \( 1 + 0.616T + 5T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 - 4.00T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 43 | \( 1 + 9.35T + 43T^{2} \) |
| 47 | \( 1 + 2.90T + 47T^{2} \) |
| 53 | \( 1 + 5.83T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 - 2.21T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.283140279617861236491241023927, −7.75408826465385295414377593726, −7.37492089584253845063195048620, −6.34493960481719307872837208161, −5.38205065719304091372522918200, −4.94773346385891147963831741159, −3.42485328310591632723444785706, −2.97449936285844218862250774659, −1.97033735192791659535094085794, −0.60158734692026153752949617451,
0.60158734692026153752949617451, 1.97033735192791659535094085794, 2.97449936285844218862250774659, 3.42485328310591632723444785706, 4.94773346385891147963831741159, 5.38205065719304091372522918200, 6.34493960481719307872837208161, 7.37492089584253845063195048620, 7.75408826465385295414377593726, 8.283140279617861236491241023927
