| L(s) = 1 | + 0.995·2-s + 0.988·3-s − 1.00·4-s − 5-s + 0.984·6-s − 2.22·7-s − 2.99·8-s − 2.02·9-s − 0.995·10-s + 1.85·11-s − 0.998·12-s + 6.23·13-s − 2.21·14-s − 0.988·15-s − 0.962·16-s − 0.366·17-s − 2.01·18-s + 8.06·19-s + 1.00·20-s − 2.20·21-s + 1.84·22-s − 6.22·23-s − 2.96·24-s + 25-s + 6.20·26-s − 4.96·27-s + 2.25·28-s + ⋯ |
| L(s) = 1 | + 0.703·2-s + 0.570·3-s − 0.504·4-s − 0.447·5-s + 0.401·6-s − 0.842·7-s − 1.05·8-s − 0.673·9-s − 0.314·10-s + 0.559·11-s − 0.288·12-s + 1.72·13-s − 0.593·14-s − 0.255·15-s − 0.240·16-s − 0.0889·17-s − 0.474·18-s + 1.84·19-s + 0.225·20-s − 0.481·21-s + 0.394·22-s − 1.29·23-s − 0.604·24-s + 0.200·25-s + 1.21·26-s − 0.955·27-s + 0.425·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.079779222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079779222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.995T + 2T^{2} \) |
| 3 | \( 1 - 0.988T + 3T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.366T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 - 8.81T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 0.306T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 3.63T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 - 8.88T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.245970104495474837554886271988, −8.376036476994176689014939217016, −7.82037721401903416781812234508, −6.46911986314523383010333197354, −6.01740931662205499780234064934, −5.09956934668226531060046628837, −3.78683511471654289363855480478, −3.64254569341203009189684767549, −2.70342296081358582449438793089, −0.855441652693674913471920077709,
0.855441652693674913471920077709, 2.70342296081358582449438793089, 3.64254569341203009189684767549, 3.78683511471654289363855480478, 5.09956934668226531060046628837, 6.01740931662205499780234064934, 6.46911986314523383010333197354, 7.82037721401903416781812234508, 8.376036476994176689014939217016, 9.245970104495474837554886271988
