| L(s) = 1 | + 2-s − 1.45·3-s + 4-s + 1.69·5-s − 1.45·6-s − 3.88·7-s + 8-s − 0.888·9-s + 1.69·10-s − 5.71·11-s − 1.45·12-s − 1.97·13-s − 3.88·14-s − 2.46·15-s + 16-s + 2.53·17-s − 0.888·18-s + 5.15·19-s + 1.69·20-s + 5.64·21-s − 5.71·22-s + 5.53·23-s − 1.45·24-s − 2.11·25-s − 1.97·26-s + 5.65·27-s − 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.838·3-s + 0.5·4-s + 0.759·5-s − 0.593·6-s − 1.46·7-s + 0.353·8-s − 0.296·9-s + 0.536·10-s − 1.72·11-s − 0.419·12-s − 0.547·13-s − 1.03·14-s − 0.636·15-s + 0.250·16-s + 0.614·17-s − 0.209·18-s + 1.18·19-s + 0.379·20-s + 1.23·21-s − 1.21·22-s + 1.15·23-s − 0.296·24-s − 0.423·25-s − 0.387·26-s + 1.08·27-s − 0.733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.361356564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361356564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 0.451T + 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 + 8.71T + 71T^{2} \) |
| 73 | \( 1 - 4.06T + 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
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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
−7.72782515735450490587542876654, −7.21917877913482695088849028490, −6.42420486492733147592888493798, −5.69665091862337811012157877974, −5.42001415461604281410142903602, −4.82242604546845997446971430526, −3.34885330933625764573361968347, −3.03339838068769499179363099172, −2.08818415812338924124138432784, −0.53868640329321644265783952192,
0.53868640329321644265783952192, 2.08818415812338924124138432784, 3.03339838068769499179363099172, 3.34885330933625764573361968347, 4.82242604546845997446971430526, 5.42001415461604281410142903602, 5.69665091862337811012157877974, 6.42420486492733147592888493798, 7.21917877913482695088849028490, 7.72782515735450490587542876654
