| L(s) = 1 | − 2.75·2-s + 2.83·3-s + 5.59·4-s − 1.76·5-s − 7.80·6-s + 2.47·7-s − 9.90·8-s + 5.01·9-s + 4.85·10-s − 11-s + 15.8·12-s − 0.921·13-s − 6.83·14-s − 4.98·15-s + 16.1·16-s + 17-s − 13.8·18-s + 1.06·19-s − 9.85·20-s + 7.01·21-s + 2.75·22-s − 6.87·23-s − 28.0·24-s − 1.89·25-s + 2.53·26-s + 5.69·27-s + 13.8·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.63·3-s + 2.79·4-s − 0.787·5-s − 3.18·6-s + 0.936·7-s − 3.50·8-s + 1.67·9-s + 1.53·10-s − 0.301·11-s + 4.57·12-s − 0.255·13-s − 1.82·14-s − 1.28·15-s + 4.02·16-s + 0.242·17-s − 3.25·18-s + 0.244·19-s − 2.20·20-s + 1.53·21-s + 0.587·22-s − 1.43·23-s − 5.72·24-s − 0.379·25-s + 0.497·26-s + 1.09·27-s + 2.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 - 2.83T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 13 | \( 1 + 0.921T + 13T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 6.87T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 + 9.01T + 41T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 9.80T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8.65T + 67T^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.88T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−7.78745066511828587294592515350, −7.46019080237056110334024433515, −6.64496810730743011854462520740, −5.62452486101578149795237507501, −4.36586721409487056328195954092, −3.50096750679036514886321735665, −2.75890642452523165159018909990, −2.00971923320540076163327729606, −1.38495827013861410823317948242, 0,
1.38495827013861410823317948242, 2.00971923320540076163327729606, 2.75890642452523165159018909990, 3.50096750679036514886321735665, 4.36586721409487056328195954092, 5.62452486101578149795237507501, 6.64496810730743011854462520740, 7.46019080237056110334024433515, 7.78745066511828587294592515350
