| L(s) = 1 | − 1.19·2-s + 3-s − 0.566·4-s − 2.06·5-s − 1.19·6-s − 5.04·7-s + 3.07·8-s + 9-s + 2.46·10-s − 3.25·11-s − 0.566·12-s − 6.23·13-s + 6.03·14-s − 2.06·15-s − 2.54·16-s − 1.68·17-s − 1.19·18-s − 6.81·19-s + 1.16·20-s − 5.04·21-s + 3.89·22-s + 23-s + 3.07·24-s − 0.747·25-s + 7.47·26-s + 27-s + 2.85·28-s + ⋯ |
| L(s) = 1 | − 0.846·2-s + 0.577·3-s − 0.283·4-s − 0.922·5-s − 0.488·6-s − 1.90·7-s + 1.08·8-s + 0.333·9-s + 0.780·10-s − 0.980·11-s − 0.163·12-s − 1.73·13-s + 1.61·14-s − 0.532·15-s − 0.636·16-s − 0.409·17-s − 0.282·18-s − 1.56·19-s + 0.261·20-s − 1.10·21-s + 0.829·22-s + 0.208·23-s + 0.627·24-s − 0.149·25-s + 1.46·26-s + 0.192·27-s + 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1426763003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1426763003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 + 3.55T + 41T^{2} \) |
| 43 | \( 1 + 5.36T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 - 4.08T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 6.20T + 89T^{2} \) |
| 97 | \( 1 + 7.98T + 97T^{2} \) |
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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
−9.244754372873517215448461359940, −8.317966370047873796207119666961, −7.84694901876753550452570921668, −7.04941304695409660910317360302, −6.34192702157040076627578441938, −4.84829310760742763177742312356, −4.21525037463276841351103841611, −3.12722733920888057683640072367, −2.35303459670690803117880725650, −0.25704280602051626612447957810,
0.25704280602051626612447957810, 2.35303459670690803117880725650, 3.12722733920888057683640072367, 4.21525037463276841351103841611, 4.84829310760742763177742312356, 6.34192702157040076627578441938, 7.04941304695409660910317360302, 7.84694901876753550452570921668, 8.317966370047873796207119666961, 9.244754372873517215448461359940
