| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s − 5·13-s + 16-s + 5·19-s + 3·22-s − 9·23-s + 5·26-s − 10·31-s − 32-s + 37-s − 5·38-s − 9·41-s − 8·43-s − 3·44-s + 9·46-s + 3·47-s − 5·52-s − 3·53-s − 12·59-s + 8·61-s + 10·62-s + 64-s − 8·67-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 1.38·13-s + 1/4·16-s + 1.14·19-s + 0.639·22-s − 1.87·23-s + 0.980·26-s − 1.79·31-s − 0.176·32-s + 0.164·37-s − 0.811·38-s − 1.40·41-s − 1.21·43-s − 0.452·44-s + 1.32·46-s + 0.437·47-s − 0.693·52-s − 0.412·53-s − 1.56·59-s + 1.02·61-s + 1.27·62-s + 1/8·64-s − 0.977·67-s + 0.712·71-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
| \(L(1)\) |
\(\approx\) |
\(0.3643778093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3643778093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.62563131503534, −15.06896498834921, −14.56632721390697, −13.90343975301128, −13.45183359083692, −12.61927503929669, −12.21241912840561, −11.72626655151099, −11.09453140391014, −10.40002403906104, −9.964056492941124, −9.568991768934005, −8.927162596560464, −8.068659783659475, −7.795832739436611, −7.214445005824457, −6.633784840808816, −5.698834570538446, −5.315640398756233, −4.606986178863088, −3.642910864758743, −2.987650696854516, −2.179869461097816, −1.617927958638775, −0.2600434679085881,
0.2600434679085881, 1.617927958638775, 2.179869461097816, 2.987650696854516, 3.642910864758743, 4.606986178863088, 5.315640398756233, 5.698834570538446, 6.633784840808816, 7.214445005824457, 7.795832739436611, 8.068659783659475, 8.927162596560464, 9.568991768934005, 9.964056492941124, 10.40002403906104, 11.09453140391014, 11.72626655151099, 12.21241912840561, 12.61927503929669, 13.45183359083692, 13.90343975301128, 14.56632721390697, 15.06896498834921, 15.62563131503534
