| L(s) = 1 | − 2-s + 4-s − 8-s − 4·13-s + 16-s − 6·17-s − 2·19-s + 4·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s − 2·37-s + 2·38-s + 6·41-s − 8·43-s + 12·47-s − 4·52-s + 6·53-s − 6·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s + 4·67-s − 6·68-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.784·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.324·38-s + 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.554·52-s + 0.824·53-s − 0.787·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s − 0.727·68-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.65176241298275, −15.42591211426041, −14.91852093257306, −14.16776846297163, −13.69412484844160, −13.08198009180332, −12.27213483887868, −12.10755530961408, −11.34429919132576, −10.67590466401460, −10.41224711882400, −9.603690285202431, −9.225383633781027, −8.513592081527192, −8.134582502357121, −7.297692581704517, −6.881985307595409, −6.303132712783811, −5.591370390880403, −4.667844368586518, −4.362642101963596, −3.275482329315603, −2.473355918316992, −2.060916204927072, −0.9121167915527328, 0,
0.9121167915527328, 2.060916204927072, 2.473355918316992, 3.275482329315603, 4.362642101963596, 4.667844368586518, 5.591370390880403, 6.303132712783811, 6.881985307595409, 7.297692581704517, 8.134582502357121, 8.513592081527192, 9.225383633781027, 9.603690285202431, 10.41224711882400, 10.67590466401460, 11.34429919132576, 12.10755530961408, 12.27213483887868, 13.08198009180332, 13.69412484844160, 14.16776846297163, 14.91852093257306, 15.42591211426041, 15.65176241298275
