| L(s) = 1 | + (1.39 + 0.297i)2-s + (0.0411 + 0.0183i)4-s + (−1.76 + 1.37i)5-s + (1.92 − 3.32i)7-s + (−2.26 − 1.64i)8-s + (−2.87 + 1.39i)10-s + (5.11 + 1.08i)11-s + (3.28 − 0.699i)13-s + (3.67 − 4.08i)14-s + (−2.73 − 3.03i)16-s + (3.90 + 2.84i)17-s + (−1.90 − 1.38i)19-s + (−0.0976 + 0.0242i)20-s + (6.82 + 3.04i)22-s + (3.12 − 3.46i)23-s + ⋯ |
| L(s) = 1 | + (0.989 + 0.210i)2-s + (0.0205 + 0.00915i)4-s + (−0.788 + 0.614i)5-s + (0.725 − 1.25i)7-s + (−0.799 − 0.580i)8-s + (−0.909 + 0.442i)10-s + (1.54 + 0.327i)11-s + (0.912 − 0.193i)13-s + (0.982 − 1.09i)14-s + (−0.683 − 0.759i)16-s + (0.948 + 0.688i)17-s + (−0.437 − 0.318i)19-s + (−0.0218 + 0.00541i)20-s + (1.45 + 0.648i)22-s + (0.650 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22417 - 0.368039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22417 - 0.368039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.76 - 1.37i)T \) |
| good | 2 | \( 1 + (-1.39 - 0.297i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.92 + 3.32i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.11 - 1.08i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.28 + 0.699i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.90 - 2.84i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.90 + 1.38i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.12 + 3.46i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.768 + 7.30i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.0679 - 0.646i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.315 + 0.970i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.66 - 0.779i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.24 - 11.8i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (2.92 - 2.12i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (7.61 - 1.61i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.44 - 1.36i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.106 + 1.01i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-2.82 + 2.04i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.38 + 4.25i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.171 + 1.63i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (10.3 - 4.59i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.935 - 2.87i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.69 + 16.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.69526026504217408869569251217, −9.703203543251170832770725715341, −8.557777918687854149582498533430, −7.65573671017314105234423491230, −6.70739256558049125784968981729, −6.08397812504707136258466016877, −4.52113732359356127808551757630, −4.11665742301732437017871380730, −3.29423473545301835096355760614, −1.09165764891312610100493095910,
1.52811628789998887093193430962, 3.27504374779281124065192190986, 3.94964996592313636325576085289, 5.09107997034882444077637360937, 5.60781015423672878931042712741, 6.82380537343167326243107680887, 8.245205774786106113416789462960, 8.762459128340429296508326427804, 9.326910675553762763679240768848, 11.07859593958428309777891779268
