L(s) = 1 | + 2-s − 2.19·3-s + 4-s − 2.75·5-s − 2.19·6-s + 8-s + 1.81·9-s − 2.75·10-s + 0.941·11-s − 2.19·12-s − 4.37·13-s + 6.04·15-s + 16-s − 1.93·17-s + 1.81·18-s + 2.67·19-s − 2.75·20-s + 0.941·22-s − 4.18·23-s − 2.19·24-s + 2.58·25-s − 4.37·26-s + 2.59·27-s − 0.145·29-s + 6.04·30-s − 7.42·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.26·3-s + 0.5·4-s − 1.23·5-s − 0.896·6-s + 0.353·8-s + 0.605·9-s − 0.870·10-s + 0.284·11-s − 0.633·12-s − 1.21·13-s + 1.56·15-s + 0.250·16-s − 0.469·17-s + 0.428·18-s + 0.614·19-s − 0.615·20-s + 0.200·22-s − 0.873·23-s − 0.448·24-s + 0.517·25-s − 0.858·26-s + 0.499·27-s − 0.0270·29-s + 1.10·30-s − 1.33·31-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
\(L(1)\) |
\(\approx\) |
\(0.7998047463\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7998047463\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 11 | \( 1 - 0.941T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 5.20T + 59T^{2} \) |
| 61 | \( 1 - 1.15T + 61T^{2} \) |
| 67 | \( 1 - 0.684T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
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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
−8.210150149528763413435835880113, −7.34212345262448183667746284028, −7.04625148861967787535845348882, −6.07033134659762394330164417761, −5.39625745478633629060154102480, −4.72601098010225317121661279320, −4.05413750999573836038343102271, −3.24069079734268779782968987534, −2.00285912517479481463877142136, −0.47061560769203629267354912918,
0.47061560769203629267354912918, 2.00285912517479481463877142136, 3.24069079734268779782968987534, 4.05413750999573836038343102271, 4.72601098010225317121661279320, 5.39625745478633629060154102480, 6.07033134659762394330164417761, 7.04625148861967787535845348882, 7.34212345262448183667746284028, 8.210150149528763413435835880113