| L(s) = 1 | − 0.568·2-s + 3-s − 1.67·4-s + 2.70·5-s − 0.568·6-s + 3.38·7-s + 2.09·8-s + 9-s − 1.53·10-s − 4.48·11-s − 1.67·12-s + 6.26·13-s − 1.92·14-s + 2.70·15-s + 2.16·16-s + 3.06·17-s − 0.568·18-s + 1.04·19-s − 4.52·20-s + 3.38·21-s + 2.55·22-s + 2.09·24-s + 2.30·25-s − 3.56·26-s + 27-s − 5.67·28-s − 2.21·29-s + ⋯ |
| L(s) = 1 | − 0.402·2-s + 0.577·3-s − 0.838·4-s + 1.20·5-s − 0.232·6-s + 1.27·7-s + 0.739·8-s + 0.333·9-s − 0.486·10-s − 1.35·11-s − 0.483·12-s + 1.73·13-s − 0.514·14-s + 0.697·15-s + 0.540·16-s + 0.744·17-s − 0.134·18-s + 0.240·19-s − 1.01·20-s + 0.738·21-s + 0.544·22-s + 0.426·24-s + 0.460·25-s − 0.698·26-s + 0.192·27-s − 1.07·28-s − 0.411·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.121467395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.121467395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.568T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 - 6.26T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 0.279T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 1.83T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 - 5.39T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 4.68T + 61T^{2} \) |
| 67 | \( 1 + 0.201T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 0.873T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.220805954706281404168150657705, −8.715121760365091257588913731665, −7.982229516388109774005724646349, −7.43965414043476729429173837731, −5.83002755378234211194352603315, −5.43127861152902909447163974896, −4.45806149524238148064360631784, −3.37679908700225275194005247321, −2.03047225702332060674570037545, −1.21246201841743233179418113235,
1.21246201841743233179418113235, 2.03047225702332060674570037545, 3.37679908700225275194005247321, 4.45806149524238148064360631784, 5.43127861152902909447163974896, 5.83002755378234211194352603315, 7.43965414043476729429173837731, 7.982229516388109774005724646349, 8.715121760365091257588913731665, 9.220805954706281404168150657705
