| L(s) = 1 | + 1.06·2-s − 3-s − 0.860·4-s + 0.994·5-s − 1.06·6-s + 7-s − 3.05·8-s + 9-s + 1.06·10-s + 0.249·11-s + 0.860·12-s + 1.06·14-s − 0.994·15-s − 1.53·16-s + 6.20·17-s + 1.06·18-s − 8.40·19-s − 0.855·20-s − 21-s + 0.266·22-s − 1.81·23-s + 3.05·24-s − 4.01·25-s − 27-s − 0.860·28-s + 2.69·29-s − 1.06·30-s + ⋯ |
| L(s) = 1 | + 0.754·2-s − 0.577·3-s − 0.430·4-s + 0.444·5-s − 0.435·6-s + 0.377·7-s − 1.07·8-s + 0.333·9-s + 0.335·10-s + 0.0751·11-s + 0.248·12-s + 0.285·14-s − 0.256·15-s − 0.384·16-s + 1.50·17-s + 0.251·18-s − 1.92·19-s − 0.191·20-s − 0.218·21-s + 0.0567·22-s − 0.378·23-s + 0.623·24-s − 0.802·25-s − 0.192·27-s − 0.162·28-s + 0.499·29-s − 0.193·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 5 | \( 1 - 0.994T + 5T^{2} \) |
| 11 | \( 1 - 0.249T + 11T^{2} \) |
| 17 | \( 1 - 6.20T + 17T^{2} \) |
| 19 | \( 1 + 8.40T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 + 8.53T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.601T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 - 6.62T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 9.78T + 89T^{2} \) |
| 97 | \( 1 + 2.43T + 97T^{2} \) |
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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
−8.215937256327682991410880147378, −7.38038988241639342007147406929, −6.32206219640752673224547557071, −5.81003751185201975876428803973, −5.25289454762457315897739404812, −4.32600866782537670962595930179, −3.81655276836371771888195840811, −2.62825718305589935023234217827, −1.48156016532944560422303383051, 0,
1.48156016532944560422303383051, 2.62825718305589935023234217827, 3.81655276836371771888195840811, 4.32600866782537670962595930179, 5.25289454762457315897739404812, 5.81003751185201975876428803973, 6.32206219640752673224547557071, 7.38038988241639342007147406929, 8.215937256327682991410880147378
