| L(s) = 1 | − 0.900·2-s − 3-s − 1.18·4-s − 5-s + 0.900·6-s + 3.65·7-s + 2.87·8-s + 9-s + 0.900·10-s − 2.25·11-s + 1.18·12-s − 1.48·13-s − 3.29·14-s + 15-s − 0.209·16-s + 7.57·17-s − 0.900·18-s + 0.0399·19-s + 1.18·20-s − 3.65·21-s + 2.02·22-s + 1.80·23-s − 2.87·24-s + 25-s + 1.33·26-s − 27-s − 4.34·28-s + ⋯ |
| L(s) = 1 | − 0.636·2-s − 0.577·3-s − 0.594·4-s − 0.447·5-s + 0.367·6-s + 1.38·7-s + 1.01·8-s + 0.333·9-s + 0.284·10-s − 0.679·11-s + 0.343·12-s − 0.411·13-s − 0.880·14-s + 0.258·15-s − 0.0524·16-s + 1.83·17-s − 0.212·18-s + 0.00916·19-s + 0.265·20-s − 0.797·21-s + 0.432·22-s + 0.376·23-s − 0.586·24-s + 0.200·25-s + 0.262·26-s − 0.192·27-s − 0.821·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.089130780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089130780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.900T + 2T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 7.57T + 17T^{2} \) |
| 19 | \( 1 - 0.0399T + 19T^{2} \) |
| 23 | \( 1 - 1.80T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 - 9.75T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + 0.240T + 41T^{2} \) |
| 43 | \( 1 - 0.947T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 + 1.72T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 0.420T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
| 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
−8.094391725869084757136493324677, −7.62948319606572761812279486195, −6.95142924437825913727746597850, −5.69150055152210415647634089245, −5.13208215556722315684965850571, −4.64810986478575637273916360042, −3.84391364827703956701396858011, −2.66885284053230513215385447661, −1.39059627997108993600258335721, −0.71539894274193258378236567523,
0.71539894274193258378236567523, 1.39059627997108993600258335721, 2.66885284053230513215385447661, 3.84391364827703956701396858011, 4.64810986478575637273916360042, 5.13208215556722315684965850571, 5.69150055152210415647634089245, 6.95142924437825913727746597850, 7.62948319606572761812279486195, 8.094391725869084757136493324677
