| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 11-s − 12-s + 13-s + 16-s − 17-s + 22-s − 2·23-s − 24-s + 25-s + 26-s + 27-s + 2·31-s + 32-s − 33-s − 34-s − 39-s + 44-s − 2·46-s − 48-s + 50-s + 51-s + 52-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 11-s − 12-s + 13-s + 16-s − 17-s + 22-s − 2·23-s − 24-s + 25-s + 26-s + 27-s + 2·31-s + 32-s − 33-s − 34-s − 39-s + 44-s − 2·46-s − 48-s + 50-s + 51-s + 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.869065267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869065267\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−8.639217012156482212330889496011, −8.016935336854337869262722094686, −6.78709060079985588274650330550, −6.33961342435495727735185602483, −5.97150607577023625911305472854, −4.92752193569597296739862696670, −4.31250544638966578912184895517, −3.51404720931635747312251876819, −2.37798867274530376125584526971, −1.18397945750732845569514875856,
1.18397945750732845569514875856, 2.37798867274530376125584526971, 3.51404720931635747312251876819, 4.31250544638966578912184895517, 4.92752193569597296739862696670, 5.97150607577023625911305472854, 6.33961342435495727735185602483, 6.78709060079985588274650330550, 8.016935336854337869262722094686, 8.639217012156482212330889496011
