| L(s) = 1 | − 2.24·2-s − 3-s + 3.03·4-s − 1.31·5-s + 2.24·6-s − 2.96·7-s − 2.32·8-s + 9-s + 2.94·10-s + 1.51·11-s − 3.03·12-s + 0.902·13-s + 6.65·14-s + 1.31·15-s − 0.850·16-s + 3.54·17-s − 2.24·18-s + 0.438·19-s − 3.98·20-s + 2.96·21-s − 3.40·22-s − 23-s + 2.32·24-s − 3.27·25-s − 2.02·26-s − 27-s − 9.01·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.577·3-s + 1.51·4-s − 0.587·5-s + 0.916·6-s − 1.12·7-s − 0.822·8-s + 0.333·9-s + 0.932·10-s + 0.457·11-s − 0.876·12-s + 0.250·13-s + 1.77·14-s + 0.339·15-s − 0.212·16-s + 0.859·17-s − 0.528·18-s + 0.100·19-s − 0.892·20-s + 0.647·21-s − 0.725·22-s − 0.208·23-s + 0.475·24-s − 0.654·25-s − 0.397·26-s − 0.192·27-s − 1.70·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3369661980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3369661980\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 0.902T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 0.438T + 19T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 2.88T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 + 7.47T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−9.257060408127468760696124323422, −8.439740734596613597583928279005, −7.70215018545302607603447848400, −6.94761601495988825344866993699, −6.37957225717903213797687460758, −5.39831832274803782383162407404, −4.05476227073023190099886421496, −3.17541719685317247030340248545, −1.71251702524780496166053823314, −0.50742962441530302429098730519,
0.50742962441530302429098730519, 1.71251702524780496166053823314, 3.17541719685317247030340248545, 4.05476227073023190099886421496, 5.39831832274803782383162407404, 6.37957225717903213797687460758, 6.94761601495988825344866993699, 7.70215018545302607603447848400, 8.439740734596613597583928279005, 9.257060408127468760696124323422
