| L(s) = 1 | − 2.05·3-s − 1.22·5-s + 1.00·7-s + 1.23·9-s − 1.98·11-s − 6.56·13-s + 2.52·15-s − 4.79·17-s + 4.50·19-s − 2.05·21-s − 7.49·23-s − 3.49·25-s + 3.63·27-s − 1.71·29-s − 6.65·31-s + 4.08·33-s − 1.22·35-s + 8.10·37-s + 13.5·39-s + 9.42·41-s − 1.99·43-s − 1.51·45-s − 3.28·47-s − 5.99·49-s + 9.87·51-s + 10.2·53-s + 2.43·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s − 0.549·5-s + 0.378·7-s + 0.411·9-s − 0.598·11-s − 1.82·13-s + 0.652·15-s − 1.16·17-s + 1.03·19-s − 0.449·21-s − 1.56·23-s − 0.698·25-s + 0.699·27-s − 0.318·29-s − 1.19·31-s + 0.710·33-s − 0.207·35-s + 1.33·37-s + 2.16·39-s + 1.47·41-s − 0.304·43-s − 0.225·45-s − 0.479·47-s − 0.857·49-s + 1.38·51-s + 1.41·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3078147713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3078147713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 - 9.42T + 41T^{2} \) |
| 43 | \( 1 + 1.99T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 + 6.27T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.211309979063701711639539469388, −7.57197975393029781532295645978, −7.11849812885795501531064164726, −6.02092062286708321018917844807, −5.52786121902144455696978601953, −4.69579464160862842980234437688, −4.20955202964647419682605999524, −2.86372789986731582427028363198, −1.93659400208522176536152189310, −0.31875170401162180542218816854,
0.31875170401162180542218816854, 1.93659400208522176536152189310, 2.86372789986731582427028363198, 4.20955202964647419682605999524, 4.69579464160862842980234437688, 5.52786121902144455696978601953, 6.02092062286708321018917844807, 7.11849812885795501531064164726, 7.57197975393029781532295645978, 8.211309979063701711639539469388
