| L(s) = 1 | − 2.35·2-s + 0.502·3-s + 3.52·4-s + 5-s − 1.18·6-s + 0.984·7-s − 3.58·8-s − 2.74·9-s − 2.35·10-s − 11-s + 1.76·12-s + 0.863·13-s − 2.31·14-s + 0.502·15-s + 1.37·16-s + 5.30·17-s + 6.45·18-s + 7.46·19-s + 3.52·20-s + 0.494·21-s + 2.35·22-s + 8.51·23-s − 1.79·24-s + 25-s − 2.02·26-s − 2.88·27-s + 3.47·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.289·3-s + 1.76·4-s + 0.447·5-s − 0.481·6-s + 0.372·7-s − 1.26·8-s − 0.915·9-s − 0.743·10-s − 0.301·11-s + 0.510·12-s + 0.239·13-s − 0.618·14-s + 0.129·15-s + 0.342·16-s + 1.28·17-s + 1.52·18-s + 1.71·19-s + 0.788·20-s + 0.107·21-s + 0.501·22-s + 1.77·23-s − 0.367·24-s + 0.200·25-s − 0.397·26-s − 0.555·27-s + 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122068518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122068518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 0.502T + 3T^{2} \) |
| 7 | \( 1 - 0.984T + 7T^{2} \) |
| 13 | \( 1 - 0.863T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 5.36T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 4.22T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 2.20T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 79 | \( 1 + 6.26T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.355508003188359112682650779126, −8.127196060239751346214258400246, −7.18219331316279290533097858413, −6.62990680283601093498657518056, −5.48101938262345095169467485897, −5.03533413770012726803407019394, −3.27236785544115981511818421074, −2.81369304711119376208985245051, −1.59074334381936794937057859694, −0.823234798658428896531075142014,
0.823234798658428896531075142014, 1.59074334381936794937057859694, 2.81369304711119376208985245051, 3.27236785544115981511818421074, 5.03533413770012726803407019394, 5.48101938262345095169467485897, 6.62990680283601093498657518056, 7.18219331316279290533097858413, 8.127196060239751346214258400246, 8.355508003188359112682650779126
