| L(s) = 1 | − 3-s + 4·7-s + 9-s − 4·11-s − 13-s − 6·17-s − 4·21-s − 4·23-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 6·51-s + 2·53-s − 4·59-s + 14·61-s + 4·63-s − 12·67-s + 4·69-s + 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s − 0.520·59-s + 1.79·61-s + 0.503·63-s − 1.46·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.07831762894163, −15.85532532410906, −15.06683541459805, −14.84872017633285, −13.91805839754785, −13.58476597497077, −12.90900076265935, −12.33296780280895, −11.61794539439106, −11.25976822527705, −10.72165843714871, −10.26958327197170, −9.482265624478791, −8.764066898141489, −8.072282832666461, −7.732125439267761, −7.057550250724624, −6.235235609616548, −5.595588634825185, −4.971033079245870, −4.505689113583437, −3.867763825407368, −2.457957069788756, −2.195296423127552, −1.081175386244187, 0,
1.081175386244187, 2.195296423127552, 2.457957069788756, 3.867763825407368, 4.505689113583437, 4.971033079245870, 5.595588634825185, 6.235235609616548, 7.057550250724624, 7.732125439267761, 8.072282832666461, 8.764066898141489, 9.482265624478791, 10.26958327197170, 10.72165843714871, 11.25976822527705, 11.61794539439106, 12.33296780280895, 12.90900076265935, 13.58476597497077, 13.91805839754785, 14.84872017633285, 15.06683541459805, 15.85532532410906, 16.07831762894163
