| L(s) = 1 | − 1.03·2-s − 0.920·3-s − 0.939·4-s + 3.95·5-s + 0.947·6-s + 4.48·7-s + 3.02·8-s − 2.15·9-s − 4.07·10-s + 5.58·11-s + 0.864·12-s + 1.54·13-s − 4.61·14-s − 3.64·15-s − 1.23·16-s − 7.55·17-s + 2.21·18-s − 3.20·19-s − 3.71·20-s − 4.12·21-s − 5.74·22-s + 23-s − 2.78·24-s + 10.6·25-s − 1.58·26-s + 4.74·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | − 0.728·2-s − 0.531·3-s − 0.469·4-s + 1.77·5-s + 0.386·6-s + 1.69·7-s + 1.07·8-s − 0.717·9-s − 1.28·10-s + 1.68·11-s + 0.249·12-s + 0.427·13-s − 1.23·14-s − 0.940·15-s − 0.309·16-s − 1.83·17-s + 0.522·18-s − 0.735·19-s − 0.831·20-s − 0.899·21-s − 1.22·22-s + 0.208·23-s − 0.568·24-s + 2.13·25-s − 0.311·26-s + 0.912·27-s − 0.795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250305615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250305615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.03T + 2T^{2} \) |
| 3 | \( 1 + 0.920T + 3T^{2} \) |
| 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 - 0.740T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 3.13T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−10.54930679403158274662904819009, −9.496213592234020553557759204472, −8.744860182561274432761186831892, −8.453981133706646949818140029108, −6.79953690724692453705293036097, −6.09773489387473984544144188636, −5.04549365173989619255737863953, −4.35715355859985418078748556629, −2.09220105089759713345930171201, −1.27211289770000172727214958157,
1.27211289770000172727214958157, 2.09220105089759713345930171201, 4.35715355859985418078748556629, 5.04549365173989619255737863953, 6.09773489387473984544144188636, 6.79953690724692453705293036097, 8.453981133706646949818140029108, 8.744860182561274432761186831892, 9.496213592234020553557759204472, 10.54930679403158274662904819009
