| L(s) = 1 | + 1.57·2-s + 0.266·3-s + 0.485·4-s − 1.88·5-s + 0.420·6-s − 0.868·7-s − 2.38·8-s − 2.92·9-s − 2.96·10-s − 4.15·11-s + 0.129·12-s − 1.68·13-s − 1.36·14-s − 0.501·15-s − 4.73·16-s + 6.19·17-s − 4.61·18-s − 0.936·19-s − 0.914·20-s − 0.231·21-s − 6.54·22-s + 23-s − 0.636·24-s − 1.45·25-s − 2.65·26-s − 1.58·27-s − 0.421·28-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.153·3-s + 0.242·4-s − 0.841·5-s + 0.171·6-s − 0.328·7-s − 0.844·8-s − 0.976·9-s − 0.938·10-s − 1.25·11-s + 0.0373·12-s − 0.467·13-s − 0.366·14-s − 0.129·15-s − 1.18·16-s + 1.50·17-s − 1.08·18-s − 0.214·19-s − 0.204·20-s − 0.0505·21-s − 1.39·22-s + 0.208·23-s − 0.129·24-s − 0.291·25-s − 0.521·26-s − 0.304·27-s − 0.0797·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 3 | \( 1 - 0.266T + 3T^{2} \) |
| 5 | \( 1 + 1.88T + 5T^{2} \) |
| 7 | \( 1 + 0.868T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 + 0.936T + 19T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 + 2.84T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 0.878T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−10.19258017992621540155613379863, −9.177888803611102172798098220475, −8.147919117603133639772836933728, −7.54054248582911110312103729178, −6.14950990070467905490593605260, −5.40095956906414930270516502909, −4.53092499899867928999651496318, −3.34975123668845509713663744539, −2.77302545919211019973833724661, 0,
2.77302545919211019973833724661, 3.34975123668845509713663744539, 4.53092499899867928999651496318, 5.40095956906414930270516502909, 6.14950990070467905490593605260, 7.54054248582911110312103729178, 8.147919117603133639772836933728, 9.177888803611102172798098220475, 10.19258017992621540155613379863
