| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 11-s − 12-s + 13-s + 14-s + 16-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + 31-s − 32-s − 33-s + 37-s − 39-s + 41-s − 42-s + 44-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 11-s − 12-s + 13-s + 14-s + 16-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + 31-s − 32-s − 33-s + 37-s − 39-s + 41-s − 42-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4215075552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4215075552\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.64395295727242441707098717042, −9.746606008942816388933381548676, −9.015583218332226410051730892396, −8.170142547218155842603136854742, −6.89413029194661086740361443487, −6.29983339380796249598384662619, −5.74085531224855117709731987684, −4.06543122933124548766140920932, −2.80969679386186843695078355233, −1.00744226478458233018468617707,
1.00744226478458233018468617707, 2.80969679386186843695078355233, 4.06543122933124548766140920932, 5.74085531224855117709731987684, 6.29983339380796249598384662619, 6.89413029194661086740361443487, 8.170142547218155842603136854742, 9.015583218332226410051730892396, 9.746606008942816388933381548676, 10.64395295727242441707098717042
