| L(s) = 1 | − 2.65·2-s + 0.381·3-s + 5.07·4-s + 4.30·5-s − 1.01·6-s + 0.973·7-s − 8.17·8-s − 2.85·9-s − 11.4·10-s − 3.68·11-s + 1.93·12-s − 5.03·13-s − 2.58·14-s + 1.64·15-s + 11.6·16-s − 1.29·17-s + 7.59·18-s − 2.92·19-s + 21.8·20-s + 0.371·21-s + 9.80·22-s + 1.79·23-s − 3.12·24-s + 13.5·25-s + 13.3·26-s − 2.23·27-s + 4.93·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.220·3-s + 2.53·4-s + 1.92·5-s − 0.414·6-s + 0.367·7-s − 2.89·8-s − 0.951·9-s − 3.62·10-s − 1.11·11-s + 0.559·12-s − 1.39·13-s − 0.691·14-s + 0.424·15-s + 2.90·16-s − 0.313·17-s + 1.78·18-s − 0.672·19-s + 4.89·20-s + 0.0810·21-s + 2.08·22-s + 0.374·23-s − 0.637·24-s + 2.71·25-s + 2.62·26-s − 0.430·27-s + 0.933·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9512947904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9512947904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 - 0.973T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 0.755T + 31T^{2} \) |
| 37 | \( 1 - 0.566T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 16.7T + 73T^{2} \) |
| 79 | \( 1 - 9.71T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.501525456210347613112259897751, −7.71628489443061315198044391195, −6.97131472477071029461011244356, −6.29810174061330567686323483236, −5.54213737455822476831819504112, −4.96726125792450021507732707354, −2.93154616710807200185139713845, −2.32533528274937127000383705728, −2.02605141022599596021108340996, −0.65948881607376489154350748523,
0.65948881607376489154350748523, 2.02605141022599596021108340996, 2.32533528274937127000383705728, 2.93154616710807200185139713845, 4.96726125792450021507732707354, 5.54213737455822476831819504112, 6.29810174061330567686323483236, 6.97131472477071029461011244356, 7.71628489443061315198044391195, 8.501525456210347613112259897751
