| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 6·13-s + 16-s + 4·19-s + 6·22-s − 6·26-s + 8·29-s − 2·31-s − 32-s − 4·37-s − 4·38-s − 10·41-s + 6·43-s − 6·44-s + 2·47-s + 6·52-s + 10·53-s − 8·58-s − 4·59-s − 14·61-s + 2·62-s + 64-s − 14·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1.66·13-s + 1/4·16-s + 0.917·19-s + 1.27·22-s − 1.17·26-s + 1.48·29-s − 0.359·31-s − 0.176·32-s − 0.657·37-s − 0.648·38-s − 1.56·41-s + 0.914·43-s − 0.904·44-s + 0.291·47-s + 0.832·52-s + 1.37·53-s − 1.05·58-s − 0.520·59-s − 1.79·61-s + 0.254·62-s + 1/8·64-s − 1.71·67-s − 0.949·71-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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.87179989603248, −15.42390615032373, −15.04296497623272, −13.91324937750001, −13.68819246681178, −13.22570652458326, −12.39814427493710, −12.00872634096194, −11.23488792268539, −10.70892549527578, −10.38367870088296, −9.850901601927746, −8.960910116401266, −8.592281055599485, −8.043477082936743, −7.480247478913633, −6.902986811472414, −6.081651583686973, −5.616184536513637, −4.967315970507476, −4.114293163654586, −3.138314925784790, −2.842875810741649, −1.772963830948985, −1.031772861700630, 0,
1.031772861700630, 1.772963830948985, 2.842875810741649, 3.138314925784790, 4.114293163654586, 4.967315970507476, 5.616184536513637, 6.081651583686973, 6.902986811472414, 7.480247478913633, 8.043477082936743, 8.592281055599485, 8.960910116401266, 9.850901601927746, 10.38367870088296, 10.70892549527578, 11.23488792268539, 12.00872634096194, 12.39814427493710, 13.22570652458326, 13.68819246681178, 13.91324937750001, 15.04296497623272, 15.42390615032373, 15.87179989603248
