| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 12-s − 13-s − 14-s + 15-s + 16-s − 2·17-s + 18-s + 7·19-s − 20-s + 21-s − 4·23-s − 24-s + 25-s − 26-s − 27-s − 28-s − 2·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.60·19-s − 0.223·20-s + 0.218·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.184059420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184059420\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.55000995721615, −14.06701618829119, −13.53812647507238, −13.00453987906181, −12.36116005892755, −12.23224895250333, −11.49663831393848, −11.09593719871193, −10.70431527457524, −9.865614657815343, −9.504512916796568, −8.935257552071222, −7.956084705984841, −7.587693985699408, −7.111633167079071, −6.475566213096150, −5.713902101327640, −5.622234661432158, −4.634207568813613, −4.305842996798891, −3.591851613048904, −2.979462696634777, −2.276521878684065, −1.362814521213105, −0.4979501585010251,
0.4979501585010251, 1.362814521213105, 2.276521878684065, 2.979462696634777, 3.591851613048904, 4.305842996798891, 4.634207568813613, 5.622234661432158, 5.713902101327640, 6.475566213096150, 7.111633167079071, 7.587693985699408, 7.956084705984841, 8.935257552071222, 9.504512916796568, 9.865614657815343, 10.70431527457524, 11.09593719871193, 11.49663831393848, 12.23224895250333, 12.36116005892755, 13.00453987906181, 13.53812647507238, 14.06701618829119, 14.55000995721615
