| L(s) = 1 | − 3-s − 2·7-s + 9-s − 13-s + 17-s − 8·19-s + 2·21-s − 6·23-s − 27-s − 7·29-s − 3·31-s − 2·37-s + 39-s − 12·41-s + 8·43-s + 6·47-s − 3·49-s − 51-s − 9·53-s + 8·57-s − 9·59-s + 2·61-s − 2·63-s − 67-s + 6·69-s − 2·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.242·17-s − 1.83·19-s + 0.436·21-s − 1.25·23-s − 0.192·27-s − 1.29·29-s − 0.538·31-s − 0.328·37-s + 0.160·39-s − 1.87·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.140·51-s − 1.23·53-s + 1.05·57-s − 1.17·59-s + 0.256·61-s − 0.251·63-s − 0.122·67-s + 0.722·69-s − 0.237·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.89494425401081, −12.59799555362577, −12.16681790740401, −11.71948270358260, −11.11114183725843, −10.72129627279054, −10.30909968034700, −9.882948177276756, −9.333867574429802, −8.979050751013320, −8.330085290606574, −7.862540906308880, −7.350695470182049, −6.796411115009982, −6.360781083619231, −5.965656319849705, −5.511234320533689, −4.892710771286111, −4.291354114396465, −3.885181956420096, −3.350055266962804, −2.667237743166936, −1.895624588610043, −1.666329456171202, −0.4696156405266651, 0,
0.4696156405266651, 1.666329456171202, 1.895624588610043, 2.667237743166936, 3.350055266962804, 3.885181956420096, 4.291354114396465, 4.892710771286111, 5.511234320533689, 5.965656319849705, 6.360781083619231, 6.796411115009982, 7.350695470182049, 7.862540906308880, 8.330085290606574, 8.979050751013320, 9.333867574429802, 9.882948177276756, 10.30909968034700, 10.72129627279054, 11.11114183725843, 11.71948270358260, 12.16681790740401, 12.59799555362577, 12.89494425401081
