| L(s) = 1 | − 3-s + 9-s + 2·11-s + 13-s + 17-s − 8·19-s − 6·23-s − 27-s − 4·31-s − 2·33-s − 2·37-s − 39-s + 4·41-s + 4·43-s − 7·49-s − 51-s + 6·53-s + 8·57-s + 4·59-s + 8·61-s − 8·67-s + 6·69-s − 4·71-s + 10·79-s + 81-s − 8·83-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 1.83·19-s − 1.25·23-s − 0.192·27-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.160·39-s + 0.624·41-s + 0.609·43-s − 49-s − 0.140·51-s + 0.824·53-s + 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.977·67-s + 0.722·69-s − 0.474·71-s + 1.12·79-s + 1/9·81-s − 0.878·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.66132651402363, −13.08352308535044, −12.75786757710981, −12.19487485644335, −11.86975765114323, −11.23140628566017, −10.89715913528261, −10.33728778204990, −9.986556711092199, −9.314862911030948, −8.862445018082427, −8.335543345265633, −7.848505096091829, −7.235490791729747, −6.614377595516836, −6.316321611523427, −5.744056253550247, −5.310197277864768, −4.481087012896751, −4.088717010773028, −3.709415066063475, −2.824845229653592, −2.059629647406056, −1.649780227810754, −0.7432080255562690, 0,
0.7432080255562690, 1.649780227810754, 2.059629647406056, 2.824845229653592, 3.709415066063475, 4.088717010773028, 4.481087012896751, 5.310197277864768, 5.744056253550247, 6.316321611523427, 6.614377595516836, 7.235490791729747, 7.848505096091829, 8.335543345265633, 8.862445018082427, 9.314862911030948, 9.986556711092199, 10.33728778204990, 10.89715913528261, 11.23140628566017, 11.86975765114323, 12.19487485644335, 12.75786757710981, 13.08352308535044, 13.66132651402363
