L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s + 58.2·7-s − 64·8-s + 81·9-s − 100·10-s − 506.·11-s − 144·12-s − 993.·13-s − 233.·14-s − 225·15-s + 256·16-s + 52.7·17-s − 324·18-s − 361·19-s + 400·20-s − 524.·21-s + 2.02e3·22-s − 1.16e3·23-s + 576·24-s + 625·25-s + 3.97e3·26-s − 729·27-s + 932.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.449·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.26·11-s − 0.288·12-s − 1.63·13-s − 0.317·14-s − 0.258·15-s + 0.250·16-s + 0.0442·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.259·21-s + 0.892·22-s − 0.460·23-s + 0.204·24-s + 0.200·25-s + 1.15·26-s − 0.192·27-s + 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8026686309\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8026686309\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
good | 7 | \( 1 - 58.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 506.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 993.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 52.7T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.71e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 417.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5T + 8.58e9T^{2} \) |
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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
−10.18395720440322267154093982602, −9.214645845462536816303644846377, −8.059815090782097304357053767339, −7.43961820095051685202411467210, −6.40816916678243770150109207291, −5.35291944217097887317337918275, −4.64437239316148803119666454678, −2.80120632721260748974990585466, −1.90616241270515794022777344932, −0.48497045696395174593120909343,
0.48497045696395174593120909343, 1.90616241270515794022777344932, 2.80120632721260748974990585466, 4.64437239316148803119666454678, 5.35291944217097887317337918275, 6.40816916678243770150109207291, 7.43961820095051685202411467210, 8.059815090782097304357053767339, 9.214645845462536816303644846377, 10.18395720440322267154093982602