| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 16-s − 18-s + 23-s + 24-s + 25-s − 27-s + 2·29-s − 32-s + 36-s − 46-s + 2·47-s − 48-s − 49-s − 50-s + 54-s − 2·58-s + 64-s − 69-s − 2·71-s − 72-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 16-s − 18-s + 23-s + 24-s + 25-s − 27-s + 2·29-s − 32-s + 36-s − 46-s + 2·47-s − 48-s − 49-s − 50-s + 54-s − 2·58-s + 64-s − 69-s − 2·71-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4560440923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4560440923\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−10.75091882471806419140566050178, −10.33923236610050813172617559681, −9.303154971195670158313161383781, −8.440278877242087088174069161239, −7.29117891204963916798434370834, −6.64783477405386543300744422844, −5.69337084403383436470517324380, −4.55674081572139923365733065023, −2.87517283825771014948491173628, −1.17102607145196359609811718938,
1.17102607145196359609811718938, 2.87517283825771014948491173628, 4.55674081572139923365733065023, 5.69337084403383436470517324380, 6.64783477405386543300744422844, 7.29117891204963916798434370834, 8.440278877242087088174069161239, 9.303154971195670158313161383781, 10.33923236610050813172617559681, 10.75091882471806419140566050178
