| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 14.2·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 28.4·10-s + 33.1·11-s − 12·12-s + 13·13-s − 14·14-s + 42.6·15-s + 16·16-s − 84.0·17-s − 18·18-s − 89.9·19-s − 56.8·20-s − 21·21-s − 66.3·22-s − 191.·23-s + 24·24-s + 77.1·25-s − 26·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.27·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.899·10-s + 0.909·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.734·15-s + 0.250·16-s − 1.19·17-s − 0.235·18-s − 1.08·19-s − 0.635·20-s − 0.218·21-s − 0.643·22-s − 1.73·23-s + 0.204·24-s + 0.617·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6322431106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6322431106\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 14.2T + 125T^{2} \) |
| 11 | \( 1 - 33.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 84.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 59.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 522.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 12.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 29.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 104.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 459.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 961.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 26.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 690.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 53.7T + 9.12e5T^{2} \) |
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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
−10.67167973011090935059159914247, −9.439642101131864647517908122210, −8.495383539084339892197453012517, −7.86896969397478682937142028017, −6.81412134708706035088197729227, −6.10223551042681472035087004775, −4.50418831563767949085190291090, −3.84094927669916456093984937943, −2.03429031494855937010414066743, −0.54372924114894329215118589321,
0.54372924114894329215118589321, 2.03429031494855937010414066743, 3.84094927669916456093984937943, 4.50418831563767949085190291090, 6.10223551042681472035087004775, 6.81412134708706035088197729227, 7.86896969397478682937142028017, 8.495383539084339892197453012517, 9.439642101131864647517908122210, 10.67167973011090935059159914247
