L(s) = 1 | + (1.23 + 0.711i)2-s + (−2.52 + 1.45i)3-s + (0.0113 + 0.0197i)4-s − 4.14·6-s + (−4.52 + 2.61i)7-s − 2.81i·8-s + (2.75 − 4.76i)9-s − 2.54·11-s + (−0.0574 − 0.0331i)12-s + (1.78 − 1.03i)13-s − 7.42·14-s + (2.02 − 3.50i)16-s + (6.55 + 3.78i)17-s + (6.77 − 3.91i)18-s + (−1.09 − 1.89i)19-s + ⋯ |
L(s) = 1 | + (0.870 + 0.502i)2-s + (−1.45 + 0.841i)3-s + (0.00569 + 0.00985i)4-s − 1.69·6-s + (−1.70 + 0.986i)7-s − 0.994i·8-s + (0.917 − 1.58i)9-s − 0.766·11-s + (−0.0165 − 0.00958i)12-s + (0.495 − 0.286i)13-s − 1.98·14-s + (0.505 − 0.875i)16-s + (1.59 + 0.918i)17-s + (1.59 − 0.922i)18-s + (−0.251 − 0.435i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.762729 - 0.111436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.762729 - 0.111436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 + (6.08 - 0.160i)T \) |
good | 2 | \( 1 + (-1.23 - 0.711i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (2.52 - 1.45i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (4.52 - 2.61i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + (-1.78 + 1.03i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-6.55 - 3.78i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.86iT - 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 41 | \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 0.751iT - 43T^{2} \) |
| 47 | \( 1 + 0.0613iT - 47T^{2} \) |
| 53 | \( 1 + (1.68 + 0.973i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.63 - 2.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.42 + 5.43i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.10 - 1.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.02iT - 73T^{2} \) |
| 79 | \( 1 + (-3.42 - 5.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.41 + 4.27i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.806 + 1.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.26iT - 97T^{2} \) |
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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
−10.20190803879406829333479957659, −9.578013180059927163116112034564, −8.442174351144642832857914556355, −6.78909957498228630156210824689, −6.26412199591866488601592384775, −5.60459898795533969533171711038, −5.13417132865368791052498052057, −3.93573985699666105335685689744, −3.10631958247699515691156179529, −0.40484607002151862863337799284,
1.06629941690554811164862224993, 2.88441005832411008950295722390, 3.74964878307923727068038455127, 4.94807709217506836917665546270, 5.72481391610090840759660448287, 6.50153240673326810729046986132, 7.28238881322129740051762040219, 8.118685738020554254005713409895, 9.757956151237100572017528226881, 10.32219869228765413240457848569