| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s + 6·13-s + 6·17-s − 2·21-s + 4·23-s − 5·25-s − 27-s + 4·29-s + 10·31-s + 4·33-s + 2·37-s − 6·39-s − 2·41-s + 8·43-s − 12·47-s − 3·49-s − 6·51-s − 12·53-s − 4·59-s + 2·61-s + 2·63-s + 4·67-s − 4·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.436·21-s + 0.834·23-s − 25-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.481·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265414252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265414252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.26026602445558485918915487570, −10.59394573181229341824784268538, −9.710853337608498914769336043468, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −6.40139715881748782453956361829, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −3.12662782741322079682173901720, −1.27109672072212401682622254955,
1.27109672072212401682622254955, 3.12662782741322079682173901720, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 6.40139715881748782453956361829, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 9.710853337608498914769336043468, 10.59394573181229341824784268538, 11.26026602445558485918915487570
