| L(s) = 1 | − 3-s − 4·5-s − 4·7-s + 9-s + 13-s + 4·15-s + 17-s − 5·19-s + 4·21-s + 4·23-s + 11·25-s − 27-s − 4·29-s − 2·31-s + 16·35-s − 6·37-s − 39-s + 43-s − 4·45-s + 9·47-s + 9·49-s − 51-s + 6·53-s + 5·57-s + 8·59-s + 6·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 1.03·15-s + 0.242·17-s − 1.14·19-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s − 0.742·29-s − 0.359·31-s + 2.70·35-s − 0.986·37-s − 0.160·39-s + 0.152·43-s − 0.596·45-s + 1.31·47-s + 9/7·49-s − 0.140·51-s + 0.824·53-s + 0.662·57-s + 1.04·59-s + 0.768·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9134932331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9134932331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.66042971405399, −12.96820587488265, −12.68021754093272, −12.43343753296548, −11.80680785000474, −11.34196244394013, −10.85824584186092, −10.51680746962944, −9.864932924817546, −9.288728923217022, −8.711850407255001, −8.338593712104771, −7.645331541708076, −7.034606562878592, −6.853198934968029, −6.294358468599315, −5.526061324430792, −5.094761826791692, −4.216834800435719, −3.832605389931318, −3.515220234879069, −2.810432136570831, −2.023477727693712, −0.7276578954490201, −0.4879271222152715,
0.4879271222152715, 0.7276578954490201, 2.023477727693712, 2.810432136570831, 3.515220234879069, 3.832605389931318, 4.216834800435719, 5.094761826791692, 5.526061324430792, 6.294358468599315, 6.853198934968029, 7.034606562878592, 7.645331541708076, 8.338593712104771, 8.711850407255001, 9.288728923217022, 9.864932924817546, 10.51680746962944, 10.85824584186092, 11.34196244394013, 11.80680785000474, 12.43343753296548, 12.68021754093272, 12.96820587488265, 13.66042971405399
