| L(s) = 1 | − 2-s + 4-s − 8-s − 2·13-s + 16-s − 3·17-s + 8·19-s − 9·23-s + 2·26-s + 6·29-s + 5·31-s − 32-s + 3·34-s − 8·37-s − 8·38-s + 3·41-s + 10·43-s + 9·46-s − 3·47-s − 2·52-s + 6·53-s − 6·58-s − 12·59-s − 4·61-s − 5·62-s + 64-s − 2·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.554·13-s + 1/4·16-s − 0.727·17-s + 1.83·19-s − 1.87·23-s + 0.392·26-s + 1.11·29-s + 0.898·31-s − 0.176·32-s + 0.514·34-s − 1.31·37-s − 1.29·38-s + 0.468·41-s + 1.52·43-s + 1.32·46-s − 0.437·47-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 1.56·59-s − 0.512·61-s − 0.635·62-s + 1/8·64-s − 0.244·67-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
| \(L(1)\) |
\(\approx\) |
\(1.252637702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252637702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.68822174695765, −15.27143579227101, −14.29221575193003, −13.93219977381196, −13.62849353993297, −12.53862950846915, −12.15060184718702, −11.79971683252454, −11.06303790282702, −10.49925352385125, −9.903018250847937, −9.531104503176265, −8.908407666398577, −8.193592466272724, −7.761166255072556, −7.176886496417561, −6.521287722592530, −5.925357702764580, −5.220919918664352, −4.516266511175670, −3.751537038047441, −2.906520907422095, −2.305889809209739, −1.431358313395624, −0.5222581308184795,
0.5222581308184795, 1.431358313395624, 2.305889809209739, 2.906520907422095, 3.751537038047441, 4.516266511175670, 5.220919918664352, 5.925357702764580, 6.521287722592530, 7.176886496417561, 7.761166255072556, 8.193592466272724, 8.908407666398577, 9.531104503176265, 9.903018250847937, 10.49925352385125, 11.06303790282702, 11.79971683252454, 12.15060184718702, 12.53862950846915, 13.62849353993297, 13.93219977381196, 14.29221575193003, 15.27143579227101, 15.68822174695765
