L(s) = 1 | − 0.136·2-s + 2.54i·3-s − 1.98·4-s + 5-s − 0.345i·6-s − 4.55i·7-s + 0.541·8-s − 3.46·9-s − 0.136·10-s + (−3.25 + 0.634i)11-s − 5.03i·12-s + 5.85·13-s + 0.619i·14-s + 2.54i·15-s + 3.88·16-s + 1.74i·17-s + ⋯ |
L(s) = 1 | − 0.0961·2-s + 1.46i·3-s − 0.990·4-s + 0.447·5-s − 0.141i·6-s − 1.72i·7-s + 0.191·8-s − 1.15·9-s − 0.0430·10-s + (−0.981 + 0.191i)11-s − 1.45i·12-s + 1.62·13-s + 0.165i·14-s + 0.656i·15-s + 0.972·16-s + 0.424i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.265058880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.265058880\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (3.25 - 0.634i)T \) |
| 19 | \( 1 + (1.61 - 4.04i)T \) |
good | 2 | \( 1 + 0.136T + 2T^{2} \) |
| 3 | \( 1 - 2.54iT - 3T^{2} \) |
| 7 | \( 1 + 4.55iT - 7T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.74iT - 17T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 8.20iT - 31T^{2} \) |
| 37 | \( 1 - 0.954iT - 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 - 2.65iT - 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 3.33iT - 61T^{2} \) |
| 67 | \( 1 - 8.57iT - 67T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 - 1.37iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.06iT - 89T^{2} \) |
| 97 | \( 1 + 1.46iT - 97T^{2} \) |
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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
−10.25370994865537523333821491841, −9.459320039804283096928620991308, −8.591517649466173848267588957549, −7.933151814511697315603888539521, −6.65887510282832453867483960807, −5.46808619433665903604561463635, −4.74464190138815479176716274910, −3.90758982202470555087447483113, −3.41627792318788991740248203387, −1.10839904508449159457776710139,
0.77743722352270211625730884843, 2.15571756704363075237152544175, 3.01270813678149738823630732447, 4.72141925073404766405711711649, 5.84240718077812777099469782363, 6.01041671644961093807036077749, 7.35328828779081108790676994800, 8.215940646106253333957081873959, 8.873528011687132383877890796646, 9.269758154007228068477040668261