| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 13-s − 17-s − 3·19-s + 2·21-s − 4·23-s − 27-s − 7·31-s + 33-s − 37-s + 39-s + 4·41-s − 7·43-s − 10·47-s − 3·49-s + 51-s + 14·53-s + 3·57-s − 10·59-s + 5·61-s − 2·63-s + 4·69-s − 10·71-s + 16·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.25·31-s + 0.174·33-s − 0.164·37-s + 0.160·39-s + 0.624·41-s − 1.06·43-s − 1.45·47-s − 3/7·49-s + 0.140·51-s + 1.92·53-s + 0.397·57-s − 1.30·59-s + 0.640·61-s − 0.251·63-s + 0.481·69-s − 1.18·71-s + 1.87·73-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 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.08070367141265, −12.59016504440435, −11.98739142966490, −11.84807837438977, −11.10026219562208, −10.70182581818741, −10.36802034194396, −9.765867326230951, −9.459580951274313, −8.932108123972224, −8.312019666248900, −7.890436730664173, −7.316336613915604, −6.824772421434234, −6.346562765471707, −6.043318578526097, −5.336515254350951, −4.995591385592509, −4.379844986935678, −3.729874289032786, −3.432668386903021, −2.594903756889050, −2.086640319473239, −1.500592052951720, −0.5291711224656856, 0,
0.5291711224656856, 1.500592052951720, 2.086640319473239, 2.594903756889050, 3.432668386903021, 3.729874289032786, 4.379844986935678, 4.995591385592509, 5.336515254350951, 6.043318578526097, 6.346562765471707, 6.824772421434234, 7.316336613915604, 7.890436730664173, 8.312019666248900, 8.932108123972224, 9.459580951274313, 9.765867326230951, 10.36802034194396, 10.70182581818741, 11.10026219562208, 11.84807837438977, 11.98739142966490, 12.59016504440435, 13.08070367141265
