| L(s) = 1 | + 0.0681·2-s + 0.0544·3-s − 1.99·4-s − 3.91·5-s + 0.00371·6-s + 0.303·7-s − 0.272·8-s − 2.99·9-s − 0.266·10-s + 11-s − 0.108·12-s + 6.79·13-s + 0.0206·14-s − 0.213·15-s + 3.97·16-s − 6.53·17-s − 0.204·18-s + 7.01·19-s + 7.81·20-s + 0.0165·21-s + 0.0681·22-s + 3.70·23-s − 0.0148·24-s + 10.3·25-s + 0.463·26-s − 0.326·27-s − 0.604·28-s + ⋯ |
| L(s) = 1 | + 0.0482·2-s + 0.0314·3-s − 0.997·4-s − 1.75·5-s + 0.00151·6-s + 0.114·7-s − 0.0963·8-s − 0.999·9-s − 0.0844·10-s + 0.301·11-s − 0.0313·12-s + 1.88·13-s + 0.00552·14-s − 0.0550·15-s + 0.993·16-s − 1.58·17-s − 0.0481·18-s + 1.60·19-s + 1.74·20-s + 0.00360·21-s + 0.0145·22-s + 0.772·23-s − 0.00302·24-s + 2.06·25-s + 0.0909·26-s − 0.0628·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7812201460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7812201460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0681T + 2T^{2} \) |
| 3 | \( 1 - 0.0544T + 3T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 - 0.303T + 7T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 - 3.70T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 1.99T + 83T^{2} \) |
| 89 | \( 1 - 3.32T + 89T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−10.88778499394944438454070334218, −9.267399227682331612424888999223, −8.685212616083431509380782430525, −8.154921383012516048525651933214, −7.15864800451095688572972423914, −5.91878050963337907772999567450, −4.81187502324351721498350546842, −3.86773909278937003431978167508, −3.26424058614637414866662960430, −0.75719306288044654971473115627,
0.75719306288044654971473115627, 3.26424058614637414866662960430, 3.86773909278937003431978167508, 4.81187502324351721498350546842, 5.91878050963337907772999567450, 7.15864800451095688572972423914, 8.154921383012516048525651933214, 8.685212616083431509380782430525, 9.267399227682331612424888999223, 10.88778499394944438454070334218
