| L(s) = 1 | + 0.213·2-s − 7.95·4-s + 15.3·5-s + 32.3·7-s − 3.40·8-s + 3.27·10-s + 29.5·11-s + 6.92·14-s + 62.9·16-s + 78.1·17-s − 10.6·19-s − 122.·20-s + 6.32·22-s + 26.8·23-s + 110.·25-s − 257.·28-s + 190.·29-s + 128.·31-s + 40.7·32-s + 16.7·34-s + 496.·35-s + 379.·37-s − 2.27·38-s − 52.2·40-s − 464.·41-s + 322.·43-s − 235.·44-s + ⋯ |
| L(s) = 1 | + 0.0755·2-s − 0.994·4-s + 1.37·5-s + 1.74·7-s − 0.150·8-s + 0.103·10-s + 0.811·11-s + 0.132·14-s + 0.982·16-s + 1.11·17-s − 0.128·19-s − 1.36·20-s + 0.0612·22-s + 0.243·23-s + 0.882·25-s − 1.73·28-s + 1.22·29-s + 0.742·31-s + 0.224·32-s + 0.0842·34-s + 2.39·35-s + 1.68·37-s − 0.00972·38-s − 0.206·40-s − 1.76·41-s + 1.14·43-s − 0.806·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.562687966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.562687966\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.213T + 8T^{2} \) |
| 5 | \( 1 - 15.3T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 11 | \( 1 - 29.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 78.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 464.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 740.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 590.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 36.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 736.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.158997625477842219994222433925, −8.329252722739139057664825604389, −7.79046174030806708274407894108, −6.43820826532599153833083329551, −5.67797860377020575198481124880, −4.90153532496053492661102101352, −4.35537467964435227219021203253, −2.97761245250613712616401650287, −1.63142224958817948584729707366, −1.06685772448659269582131288574,
1.06685772448659269582131288574, 1.63142224958817948584729707366, 2.97761245250613712616401650287, 4.35537467964435227219021203253, 4.90153532496053492661102101352, 5.67797860377020575198481124880, 6.43820826532599153833083329551, 7.79046174030806708274407894108, 8.329252722739139057664825604389, 9.158997625477842219994222433925
