L(s) = 1 | + 3-s + 9-s + 13-s − 25-s + 27-s + 39-s − 2·43-s + 49-s + 2·61-s − 75-s − 2·79-s + 81-s + 2·103-s + 117-s + ⋯ |
L(s) = 1 | + 3-s + 9-s + 13-s − 25-s + 27-s + 39-s − 2·43-s + 49-s + 2·61-s − 75-s − 2·79-s + 81-s + 2·103-s + 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.802797522\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802797522\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 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
−8.899052294190723230875790446230, −8.459642575063722469491298314321, −7.71015096411619849619435785724, −6.92312735515726177889656130751, −6.11331329585199128535546908517, −5.11131705568643717955460499065, −4.04592713799017334169992353518, −3.47799947726139976078836509473, −2.41668333385465643697704711645, −1.40600311054493191521507126043,
1.40600311054493191521507126043, 2.41668333385465643697704711645, 3.47799947726139976078836509473, 4.04592713799017334169992353518, 5.11131705568643717955460499065, 6.11331329585199128535546908517, 6.92312735515726177889656130751, 7.71015096411619849619435785724, 8.459642575063722469491298314321, 8.899052294190723230875790446230