| L(s) = 1 | + 2·3-s + 12·5-s + 7·7-s − 23·9-s − 48·11-s − 56·13-s + 24·15-s − 114·17-s − 2·19-s + 14·21-s − 120·23-s + 19·25-s − 100·27-s + 54·29-s + 236·31-s − 96·33-s + 84·35-s − 146·37-s − 112·39-s + 126·41-s + 376·43-s − 276·45-s − 12·47-s + 49·49-s − 228·51-s − 174·53-s − 576·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.07·5-s + 0.377·7-s − 0.851·9-s − 1.31·11-s − 1.19·13-s + 0.413·15-s − 1.62·17-s − 0.0241·19-s + 0.145·21-s − 1.08·23-s + 0.151·25-s − 0.712·27-s + 0.345·29-s + 1.36·31-s − 0.506·33-s + 0.405·35-s − 0.648·37-s − 0.459·39-s + 0.479·41-s + 1.33·43-s − 0.914·45-s − 0.0372·47-s + 1/7·49-s − 0.626·51-s − 0.450·53-s − 1.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 138 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 378 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−10.17637691660809895302981680936, −9.365564238324829429241527318316, −8.429275576522712961438138932296, −7.62439168231374487098148277575, −6.34799136971023390773890336012, −5.44721152386432458707400621776, −4.51867879935786841495557541546, −2.66415363648355906437371169918, −2.16730444532053418676174383212, 0,
2.16730444532053418676174383212, 2.66415363648355906437371169918, 4.51867879935786841495557541546, 5.44721152386432458707400621776, 6.34799136971023390773890336012, 7.62439168231374487098148277575, 8.429275576522712961438138932296, 9.365564238324829429241527318316, 10.17637691660809895302981680936
