| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 2·13-s + 16-s + 4·17-s − 2·22-s + 8·23-s + 2·26-s + 2·31-s + 32-s + 4·34-s − 8·37-s − 2·41-s + 2·43-s − 2·44-s + 8·46-s − 10·47-s + 2·52-s − 2·53-s + 4·59-s + 10·61-s + 2·62-s + 64-s − 2·67-s + 4·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.426·22-s + 1.66·23-s + 0.392·26-s + 0.359·31-s + 0.176·32-s + 0.685·34-s − 1.31·37-s − 0.312·41-s + 0.304·43-s − 0.301·44-s + 1.17·46-s − 1.45·47-s + 0.277·52-s − 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.254·62-s + 1/8·64-s − 0.244·67-s + 0.485·68-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
| \(L(1)\) |
\(\approx\) |
\(3.983339330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.983339330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.43390909488395, −14.96296644867933, −14.47095636474819, −13.85253587819716, −13.35653948466609, −12.89966955868024, −12.33979900363166, −11.84099711773001, −11.04638245106021, −10.84871116092659, −10.06695724959743, −9.562147495507480, −8.762459558171668, −8.164721005802471, −7.667667995161204, −6.766863353303322, −6.604679636303829, −5.471693128799512, −5.309387542515321, −4.608252536994640, −3.659936193938271, −3.266138290196486, −2.515292461403804, −1.605035114760778, −0.7433083758122876,
0.7433083758122876, 1.605035114760778, 2.515292461403804, 3.266138290196486, 3.659936193938271, 4.608252536994640, 5.309387542515321, 5.471693128799512, 6.604679636303829, 6.766863353303322, 7.667667995161204, 8.164721005802471, 8.762459558171668, 9.562147495507480, 10.06695724959743, 10.84871116092659, 11.04638245106021, 11.84099711773001, 12.33979900363166, 12.89966955868024, 13.35653948466609, 13.85253587819716, 14.47095636474819, 14.96296644867933, 15.43390909488395
