| L(s) = 1 | − 0.0953·2-s − 2.36·3-s − 1.99·4-s − 3.81·5-s + 0.225·6-s − 2.61·7-s + 0.380·8-s + 2.58·9-s + 0.363·10-s + 2.05·11-s + 4.70·12-s − 2.14·13-s + 0.249·14-s + 9.01·15-s + 3.94·16-s + 5.20·17-s − 0.246·18-s + 0.822·19-s + 7.59·20-s + 6.19·21-s − 0.195·22-s − 2.73·23-s − 0.899·24-s + 9.54·25-s + 0.204·26-s + 0.974·27-s + 5.21·28-s + ⋯ |
| L(s) = 1 | − 0.0674·2-s − 1.36·3-s − 0.995·4-s − 1.70·5-s + 0.0920·6-s − 0.990·7-s + 0.134·8-s + 0.862·9-s + 0.115·10-s + 0.618·11-s + 1.35·12-s − 0.593·13-s + 0.0667·14-s + 2.32·15-s + 0.986·16-s + 1.26·17-s − 0.0581·18-s + 0.188·19-s + 1.69·20-s + 1.35·21-s − 0.0417·22-s − 0.569·23-s − 0.183·24-s + 1.90·25-s + 0.0400·26-s + 0.187·27-s + 0.985·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.0953T + 2T^{2} \) |
| 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 0.822T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 - 0.862T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 47 | \( 1 - 0.00879T + 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 0.230T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 0.267T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−8.857082969344702375777588862519, −7.981326132193380369420997857271, −7.27219193069135369394452246384, −6.44874406414079538943819787922, −5.49478698146522932374951972463, −4.78550375688925573708330283711, −3.88383966529493576487892991550, −3.32351307307733423693289679033, −0.851544563243426163040662232965, 0,
0.851544563243426163040662232965, 3.32351307307733423693289679033, 3.88383966529493576487892991550, 4.78550375688925573708330283711, 5.49478698146522932374951972463, 6.44874406414079538943819787922, 7.27219193069135369394452246384, 7.981326132193380369420997857271, 8.857082969344702375777588862519
