| L(s) = 1 | − 2.63·2-s − 3-s + 4.93·4-s + 2.62·5-s + 2.63·6-s + 5.18·7-s − 7.73·8-s + 9-s − 6.91·10-s − 11-s − 4.93·12-s − 3.47·13-s − 13.6·14-s − 2.62·15-s + 10.5·16-s + 1.94·17-s − 2.63·18-s − 0.343·19-s + 12.9·20-s − 5.18·21-s + 2.63·22-s − 4.48·23-s + 7.73·24-s + 1.89·25-s + 9.14·26-s − 27-s + 25.6·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.46·4-s + 1.17·5-s + 1.07·6-s + 1.96·7-s − 2.73·8-s + 0.333·9-s − 2.18·10-s − 0.301·11-s − 1.42·12-s − 0.963·13-s − 3.65·14-s − 0.677·15-s + 2.62·16-s + 0.472·17-s − 0.620·18-s − 0.0787·19-s + 2.89·20-s − 1.13·21-s + 0.561·22-s − 0.935·23-s + 1.57·24-s + 0.378·25-s + 1.79·26-s − 0.192·27-s + 4.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9364728502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9364728502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 0.343T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 + 5.47T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.436T + 41T^{2} \) |
| 43 | \( 1 - 6.40T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 67 | \( 1 - 2.76T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 6.51T + 97T^{2} \) |
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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
−9.211709608942438949009244172615, −8.404398574760976409916372370323, −7.65667332249275793831673067142, −7.26666898803046358614328687573, −5.98602942316325649509718330422, −5.54979279113633760062355393617, −4.46966330199549159719788630600, −2.39062761381165655322640567816, −1.91813115605362944917786524749, −0.908316941607675956655667604735,
0.908316941607675956655667604735, 1.91813115605362944917786524749, 2.39062761381165655322640567816, 4.46966330199549159719788630600, 5.54979279113633760062355393617, 5.98602942316325649509718330422, 7.26666898803046358614328687573, 7.65667332249275793831673067142, 8.404398574760976409916372370323, 9.211709608942438949009244172615
