| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s + 11-s + 2·15-s + 4·19-s − 8·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s − 8·31-s − 2·33-s − 4·35-s − 2·37-s − 6·41-s + 8·43-s − 45-s − 6·47-s + 9·49-s − 6·53-s − 55-s − 8·57-s + 12·59-s + 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s − 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.13182016121562, −14.55371279300187, −14.18080220633539, −13.72659400905952, −12.80669288604833, −12.35156072038881, −11.87359079189165, −11.37257800327841, −11.07398980107158, −10.75326382250250, −9.819616660965042, −9.430500672483126, −8.507081011539647, −8.155206012734284, −7.569750861834185, −7.005028434593594, −6.380408965987580, −5.549269887922471, −5.300029545341790, −4.807713325197923, −3.930040362383552, −3.592253469697242, −2.339193110648455, −1.690039587015820, −0.9183174543444516, 0,
0.9183174543444516, 1.690039587015820, 2.339193110648455, 3.592253469697242, 3.930040362383552, 4.807713325197923, 5.300029545341790, 5.549269887922471, 6.380408965987580, 7.005028434593594, 7.569750861834185, 8.155206012734284, 8.507081011539647, 9.430500672483126, 9.819616660965042, 10.75326382250250, 11.07398980107158, 11.37257800327841, 11.87359079189165, 12.35156072038881, 12.80669288604833, 13.72659400905952, 14.18080220633539, 14.55371279300187, 15.13182016121562
