| L(s) = 1 | + (−1 + 2.82i)3-s + (−7.00 − 5.65i)9-s − 14·11-s − 33.9i·17-s + 16.9i·19-s − 25·25-s + (23.0 − 14.1i)27-s + (14 − 39.5i)33-s − 67.8i·41-s − 84.8i·43-s − 49·49-s + (96 + 33.9i)51-s + (−48 − 16.9i)57-s − 82·59-s + 118. i·67-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)3-s + (−0.777 − 0.628i)9-s − 1.27·11-s − 1.99i·17-s + 0.893i·19-s − 25-s + (0.851 − 0.523i)27-s + (0.424 − 1.19i)33-s − 1.65i·41-s − 1.97i·43-s − 0.999·49-s + (1.88 + 0.665i)51-s + (−0.842 − 0.297i)57-s − 1.38·59-s + 1.77i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.203224 - 0.287402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203224 - 0.287402i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 2.82i)T \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 14T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 33.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 82T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 142T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 + 158T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−10.73426967616729772657117154096, −9.998819029082505393775656228938, −9.230670982289797891026353391341, −8.120293936511278772590669494619, −7.09025044442200834980042089307, −5.65684783431656092019290611163, −5.07074281665342037675154981021, −3.81232051520960213740770621051, −2.58210186700597502667321017953, −0.15471589191163126114192393654,
1.66206459270635103636849070426, 2.96466223133164404152645670631, 4.65851734025707899421764575577, 5.81750940026169335368597904249, 6.56317823631906136196960172286, 7.88266636554958245247308278228, 8.187346667490695667509054109090, 9.635821794474120245576439409597, 10.75024794439927071395623252280, 11.30259641724257649280160975024
