L(s) = 1 | + 0.901·2-s − 3-s − 1.18·4-s − 2.82·5-s − 0.901·6-s − 0.878·7-s − 2.87·8-s + 9-s − 2.54·10-s + 4.81·11-s + 1.18·12-s − 5.14·13-s − 0.792·14-s + 2.82·15-s − 0.216·16-s − 17-s + 0.901·18-s − 0.823·19-s + 3.35·20-s + 0.878·21-s + 4.34·22-s + 2.70·23-s + 2.87·24-s + 2.98·25-s − 4.63·26-s − 27-s + 1.04·28-s + ⋯ |
L(s) = 1 | + 0.637·2-s − 0.577·3-s − 0.593·4-s − 1.26·5-s − 0.368·6-s − 0.332·7-s − 1.01·8-s + 0.333·9-s − 0.805·10-s + 1.45·11-s + 0.342·12-s − 1.42·13-s − 0.211·14-s + 0.729·15-s − 0.0541·16-s − 0.242·17-s + 0.212·18-s − 0.189·19-s + 0.749·20-s + 0.191·21-s + 0.925·22-s + 0.564·23-s + 0.586·24-s + 0.596·25-s − 0.909·26-s − 0.192·27-s + 0.197·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 0.901T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 0.878T + 7T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 19 | \( 1 + 0.823T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 - 7.06T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 2.88T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 5.27T + 67T^{2} \) |
| 71 | \( 1 - 0.559T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 0.915T + 89T^{2} \) |
| 97 | \( 1 + 0.734T + 97T^{2} \) |
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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
−7.34308749350712298507161099778, −6.74238852303051539182742014802, −6.07724036037831842174191070366, −5.20212579141039996013808567883, −4.55197205428835466763710565813, −4.03687965058738087326630565792, −3.48784949069710902993137718037, −2.45806444107566643858309961794, −0.914030171459538142267267857182, 0,
0.914030171459538142267267857182, 2.45806444107566643858309961794, 3.48784949069710902993137718037, 4.03687965058738087326630565792, 4.55197205428835466763710565813, 5.20212579141039996013808567883, 6.07724036037831842174191070366, 6.74238852303051539182742014802, 7.34308749350712298507161099778