| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s − 2·13-s − 15-s + 16-s − 18-s − 19-s − 20-s + 8·23-s − 24-s + 25-s + 2·26-s + 27-s + 6·29-s + 30-s + 6·31-s − 32-s + 36-s + 2·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.07·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.320085157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.320085157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.13489195913724, −12.75040656975981, −12.28926309673648, −11.78892996681025, −11.30102070708199, −10.76676236172616, −10.39978421580929, −9.810181916905135, −9.377403514874540, −8.910917855679246, −8.477978479734346, −7.930976003218623, −7.615305535641841, −7.025934700186013, −6.532072018063705, −6.170525139461536, −5.122700757407258, −4.885500551482991, −4.218210745784955, −3.544717282265669, −2.886152877012249, −2.629847161097837, −1.837634484987713, −1.056793629213487, −0.5501557780544192,
0.5501557780544192, 1.056793629213487, 1.837634484987713, 2.629847161097837, 2.886152877012249, 3.544717282265669, 4.218210745784955, 4.885500551482991, 5.122700757407258, 6.170525139461536, 6.532072018063705, 7.025934700186013, 7.615305535641841, 7.930976003218623, 8.477978479734346, 8.910917855679246, 9.377403514874540, 9.810181916905135, 10.39978421580929, 10.76676236172616, 11.30102070708199, 11.78892996681025, 12.28926309673648, 12.75040656975981, 13.13489195913724
