L(s) = 1 | + 1.41·5-s − 2.82i·7-s − 4i·11-s + 2i·13-s − 1.41i·17-s + 5.65·19-s − 4·23-s − 2.99·25-s − 7.07·29-s − 8.48i·31-s − 4.00i·35-s − 8i·37-s + 4.24i·41-s + 11.3·43-s − 12·47-s + ⋯ |
L(s) = 1 | + 0.632·5-s − 1.06i·7-s − 1.20i·11-s + 0.554i·13-s − 0.342i·17-s + 1.29·19-s − 0.834·23-s − 0.599·25-s − 1.31·29-s − 1.52i·31-s − 0.676i·35-s − 1.31i·37-s + 0.662i·41-s + 1.72·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.640463434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.640463434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.636872893548240786903893041971, −8.975468562336693747804015119980, −7.77064716042609425624575829875, −7.29694276800036318382918220794, −6.06875495580316535467887166582, −5.58880334786909656160455201104, −4.24127336391368014077821104667, −3.48279827679155516330239815175, −2.12096906952456972180466639581, −0.72099986513448904985670145152,
1.65203862045949977474044749941, 2.56028467783793664656114994770, 3.77173583554050019604924978975, 5.14231027992577088835262227129, 5.58277496465225715548969234861, 6.58655322798530919482034507423, 7.54596284663447331060072698631, 8.363234136187880338928014859754, 9.400200232905074030465434889703, 9.765195474310771316929182801913