| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 25·5-s − 36·6-s − 227.·7-s + 64·8-s + 81·9-s − 100·10-s − 373.·11-s − 144·12-s + 97.5·13-s − 911.·14-s + 225·15-s + 256·16-s − 798.·17-s + 324·18-s − 361·19-s − 400·20-s + 2.05e3·21-s − 1.49e3·22-s − 3.29e3·23-s − 576·24-s + 625·25-s + 390.·26-s − 729·27-s − 3.64e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.75·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.931·11-s − 0.288·12-s + 0.160·13-s − 1.24·14-s + 0.258·15-s + 0.250·16-s − 0.670·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 1.01·21-s − 0.658·22-s − 1.29·23-s − 0.204·24-s + 0.200·25-s + 0.113·26-s − 0.192·27-s − 0.879·28-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
| \(L(3)\) |
\(\approx\) |
\(0.9218485618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9218485618\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 227.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 373.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 97.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 798.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 878.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.87e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 564.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.08e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.26e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 996.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e4T + 8.58e9T^{2} \) |
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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
−10.19374310972192431833572979351, −9.245390288702036729826749847201, −7.968719762912565933407201642162, −6.94955245719098779225984693106, −6.26306661781848836068811310521, −5.44408890885469650906373297191, −4.23816024871849850356490789822, −3.38942564153690355695302304614, −2.29614892180932482755533999292, −0.40681077686011224884294230278,
0.40681077686011224884294230278, 2.29614892180932482755533999292, 3.38942564153690355695302304614, 4.23816024871849850356490789822, 5.44408890885469650906373297191, 6.26306661781848836068811310521, 6.94955245719098779225984693106, 7.968719762912565933407201642162, 9.245390288702036729826749847201, 10.19374310972192431833572979351
