L(s) = 1 | + (0.595 − 0.595i)5-s − 1.64i·7-s + (3.36 − 3.36i)11-s + (−2.64 − 2.64i)13-s − 5.53·17-s + (3.64 + 3.64i)19-s − 4.33i·23-s + 4.29i·25-s + (−6.12 − 6.12i)29-s − 5.64·31-s + (−0.979 − 0.979i)35-s + (0.645 − 0.645i)37-s − 7.91i·41-s + (0.354 − 0.354i)43-s + 9.10·47-s + ⋯ |
L(s) = 1 | + (0.266 − 0.266i)5-s − 0.622i·7-s + (1.01 − 1.01i)11-s + (−0.733 − 0.733i)13-s − 1.34·17-s + (0.836 + 0.836i)19-s − 0.904i·23-s + 0.858i·25-s + (−1.13 − 1.13i)29-s − 1.01·31-s + (−0.165 − 0.165i)35-s + (0.106 − 0.106i)37-s − 1.23i·41-s + (0.0540 − 0.0540i)43-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.395028723\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395028723\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.595 + 0.595i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.64iT - 7T^{2} \) |
| 11 | \( 1 + (-3.36 + 3.36i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.64 + 2.64i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 + (-3.64 - 3.64i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.33iT - 23T^{2} \) |
| 29 | \( 1 + (6.12 + 6.12i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.64T + 31T^{2} \) |
| 37 | \( 1 + (-0.645 + 0.645i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.91iT - 41T^{2} \) |
| 43 | \( 1 + (-0.354 + 0.354i)T - 43iT^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + (4.93 - 4.93i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.33 + 4.33i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.645 - 0.645i)T + 61iT^{2} \) |
| 67 | \( 1 + (4 + 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 + 3.29iT - 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + (3.36 + 3.36i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.38iT - 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.373973964553987858296722698661, −8.971126332595282639296553888075, −7.84936492840325683988094169185, −7.15828654417471580618320219547, −6.12129702253951470142849216162, −5.40886405374336041860439518420, −4.22461703997868845669049183981, −3.44096332639041984133794573081, −2.03562898235546417008442286797, −0.59687165222954820344282497790,
1.72454084157478137128531840367, 2.60756449431029740257920822735, 3.99183524117120684087988882506, 4.84795671399955397449019232897, 5.81549001211945879877339160051, 7.01889074183495884206314918193, 7.11370935737466753667913081873, 8.620564199665559814175327526867, 9.390752402342570192139564704158, 9.663017023397310161030568838687