| L(s) = 1 | − 2-s − 1.71·3-s + 4-s − 1.04·5-s + 1.71·6-s + 0.0655·7-s − 8-s − 0.0503·9-s + 1.04·10-s + 4.27·11-s − 1.71·12-s − 6.72·13-s − 0.0655·14-s + 1.80·15-s + 16-s + 2.76·17-s + 0.0503·18-s − 6.40·19-s − 1.04·20-s − 0.112·21-s − 4.27·22-s + 2.26·23-s + 1.71·24-s − 3.90·25-s + 6.72·26-s + 5.23·27-s + 0.0655·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.991·3-s + 0.5·4-s − 0.468·5-s + 0.701·6-s + 0.0247·7-s − 0.353·8-s − 0.0167·9-s + 0.331·10-s + 1.28·11-s − 0.495·12-s − 1.86·13-s − 0.0175·14-s + 0.464·15-s + 0.250·16-s + 0.670·17-s + 0.0118·18-s − 1.47·19-s − 0.234·20-s − 0.0245·21-s − 0.910·22-s + 0.473·23-s + 0.350·24-s − 0.780·25-s + 1.31·26-s + 1.00·27-s + 0.0123·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3502500115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3502500115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 - 0.0655T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 6.40T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 37 | \( 1 + 9.59T + 37T^{2} \) |
| 41 | \( 1 - 7.28T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 - 4.91T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 0.131T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 3.01T + 89T^{2} \) |
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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
−7.977713061650115573759600800427, −7.33992242227325841629465111140, −6.75589446277587307318449523237, −6.04304442291958141065426523496, −5.34622855538076648995643928816, −4.50554779158125881219831189781, −3.72394021810224953381346334784, −2.59320228360929504473896417591, −1.62213641305908678632117459755, −0.36137590985585533662958876236,
0.36137590985585533662958876236, 1.62213641305908678632117459755, 2.59320228360929504473896417591, 3.72394021810224953381346334784, 4.50554779158125881219831189781, 5.34622855538076648995643928816, 6.04304442291958141065426523496, 6.75589446277587307318449523237, 7.33992242227325841629465111140, 7.977713061650115573759600800427
