| L(s) = 1 | + 2.45·2-s + 4.04·4-s − 3.33·5-s + 3.69·7-s + 5.03·8-s − 8.20·10-s + 0.270·11-s + 9.08·14-s + 4.29·16-s + 4.04·17-s + 7.15·19-s − 13.5·20-s + 0.664·22-s + 2.79·23-s + 6.13·25-s + 14.9·28-s − 7.83·29-s + 2.14·31-s + 0.487·32-s + 9.93·34-s − 12.3·35-s + 4.37·37-s + 17.6·38-s − 16.8·40-s + 7.61·41-s − 3.89·43-s + 1.09·44-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.02·4-s − 1.49·5-s + 1.39·7-s + 1.78·8-s − 2.59·10-s + 0.0815·11-s + 2.42·14-s + 1.07·16-s + 0.980·17-s + 1.64·19-s − 3.02·20-s + 0.141·22-s + 0.583·23-s + 1.22·25-s + 2.82·28-s − 1.45·29-s + 0.385·31-s + 0.0861·32-s + 1.70·34-s − 2.08·35-s + 0.719·37-s + 2.85·38-s − 2.65·40-s + 1.18·41-s − 0.593·43-s + 0.165·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.535149664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.535149664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 - 0.270T + 11T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 23 | \( 1 - 2.79T + 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 - 7.61T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 + 0.877T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 + 0.405T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.499956273555522508769255229434, −8.229414965752342767599373077042, −7.59256375736145576746044196832, −7.16291420245553017831419313675, −5.79990843119194536462415456562, −5.11460783787209177296320924148, −4.42658695830240904735448181386, −3.67040196301193871730429893843, −2.88839930842753397289667977964, −1.35607324180411067045628738134,
1.35607324180411067045628738134, 2.88839930842753397289667977964, 3.67040196301193871730429893843, 4.42658695830240904735448181386, 5.11460783787209177296320924148, 5.79990843119194536462415456562, 7.16291420245553017831419313675, 7.59256375736145576746044196832, 8.229414965752342767599373077042, 9.499956273555522508769255229434
