L(s) = 1 | + 7-s + 9-s + 11-s − 23-s − 29-s + 37-s − 43-s + 49-s − 2·53-s + 63-s − 67-s + 71-s + 77-s + 79-s + 81-s + 99-s + 2·107-s − 109-s + 113-s + ⋯ |
L(s) = 1 | + 7-s + 9-s + 11-s − 23-s − 29-s + 37-s − 43-s + 49-s − 2·53-s + 63-s − 67-s + 71-s + 77-s + 79-s + 81-s + 99-s + 2·107-s − 109-s + 113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.550216329\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.550216329\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.062867718983863692086904261148, −8.078224520505955820278357337755, −7.60556394649113562372925465858, −6.71811146852535546447912777677, −5.99674018589067699365489403996, −4.95056269776157168284528288461, −4.28729279699469238452936532867, −3.54980685407027733065850215797, −2.07460971522949452813790853645, −1.33475737253289265488402896580,
1.33475737253289265488402896580, 2.07460971522949452813790853645, 3.54980685407027733065850215797, 4.28729279699469238452936532867, 4.95056269776157168284528288461, 5.99674018589067699365489403996, 6.71811146852535546447912777677, 7.60556394649113562372925465858, 8.078224520505955820278357337755, 9.062867718983863692086904261148