| L(s) = 1 | − 5-s − 7-s + 9-s + 17-s + 23-s + 25-s − 29-s + 31-s + 35-s + 37-s − 41-s + 2·43-s − 45-s + 53-s + 59-s − 63-s − 67-s + 71-s + 81-s − 83-s − 85-s − 2·97-s − 101-s + 2·103-s − 107-s + 113-s − 115-s + ⋯ |
| L(s) = 1 | − 5-s − 7-s + 9-s + 17-s + 23-s + 25-s − 29-s + 31-s + 35-s + 37-s − 41-s + 2·43-s − 45-s + 53-s + 59-s − 63-s − 67-s + 71-s + 81-s − 83-s − 85-s − 2·97-s − 101-s + 2·103-s − 107-s + 113-s − 115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9484836594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9484836594\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−9.527480437769610808204064078973, −8.663388116802846581298223362084, −7.69425274925565433646195238516, −7.19015180011613669301157922505, −6.42065219380660280149382692539, −5.36834497066931379915657560904, −4.32523427298179347282566993893, −3.63745136853782111012616740964, −2.74452236035746829472615795306, −1.02262142981773408609355804375,
1.02262142981773408609355804375, 2.74452236035746829472615795306, 3.63745136853782111012616740964, 4.32523427298179347282566993893, 5.36834497066931379915657560904, 6.42065219380660280149382692539, 7.19015180011613669301157922505, 7.69425274925565433646195238516, 8.663388116802846581298223362084, 9.527480437769610808204064078973
