| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 2·11-s − 4·13-s + 2·15-s + 17-s − 2·19-s − 21-s + 4·23-s − 25-s + 27-s − 4·29-s − 8·31-s + 2·33-s − 2·35-s + 4·37-s − 4·39-s − 10·41-s − 8·43-s + 2·45-s + 49-s + 51-s − 6·53-s + 4·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.516·15-s + 0.242·17-s − 0.458·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s + 0.348·33-s − 0.338·35-s + 0.657·37-s − 0.640·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s − 0.824·53-s + 0.539·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−16.72962931617592, −16.37550775511462, −15.37081138600945, −14.90387964727287, −14.52737912830950, −13.93646320933658, −13.19691582092656, −12.97780534771466, −12.25148070199676, −11.58712018927877, −10.89216975775930, −10.07090311689023, −9.728663046439175, −9.232595853005995, −8.629158344225750, −7.875248119586924, −7.045204602071776, −6.757486632489143, −5.807767826109081, −5.272975252381796, −4.457283979419314, −3.615032968533114, −2.933725650517540, −2.079593728518931, −1.488000007163257, 0,
1.488000007163257, 2.079593728518931, 2.933725650517540, 3.615032968533114, 4.457283979419314, 5.272975252381796, 5.807767826109081, 6.757486632489143, 7.045204602071776, 7.875248119586924, 8.629158344225750, 9.232595853005995, 9.728663046439175, 10.07090311689023, 10.89216975775930, 11.58712018927877, 12.25148070199676, 12.97780534771466, 13.19691582092656, 13.93646320933658, 14.52737912830950, 14.90387964727287, 15.37081138600945, 16.37550775511462, 16.72962931617592
