L(s) = 1 | + (0.154 + 0.178i)2-s + (0.276 − 1.92i)4-s + (1.84 + 0.540i)5-s + (2.11 + 2.44i)7-s + (0.784 − 0.504i)8-s + (0.188 + 0.412i)10-s + (1.71 + 0.502i)11-s + (1.19 + 0.768i)13-s + (−0.108 + 0.755i)14-s + (−3.51 − 1.03i)16-s + (−0.308 − 2.14i)17-s + (−3.09 + 3.57i)19-s + (1.54 − 3.39i)20-s + (0.175 + 0.383i)22-s + (−1.86 + 4.08i)23-s + ⋯ |
L(s) = 1 | + (0.109 + 0.126i)2-s + (0.138 − 0.962i)4-s + (0.823 + 0.241i)5-s + (0.799 + 0.922i)7-s + (0.277 − 0.178i)8-s + (0.0596 + 0.130i)10-s + (0.515 + 0.151i)11-s + (0.331 + 0.213i)13-s + (−0.0290 + 0.201i)14-s + (−0.879 − 0.258i)16-s + (−0.0748 − 0.520i)17-s + (−0.710 + 0.819i)19-s + (0.346 − 0.758i)20-s + (0.0373 + 0.0817i)22-s + (−0.389 + 0.852i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04090 - 0.0120408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04090 - 0.0120408i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 67 | \( 1 + (-5.87 + 5.69i)T \) |
good | 2 | \( 1 + (-0.154 - 0.178i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-1.84 - 0.540i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.11 - 2.44i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.71 - 0.502i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.19 - 0.768i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.308 + 2.14i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (3.09 - 3.57i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.86 - 4.08i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + (-8.25 + 5.30i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + (1.31 + 9.14i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.263 + 1.83i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (2.54 - 5.57i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (0.376 - 2.62i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-9.96 + 6.40i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (8.04 - 2.36i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (1.09 - 7.58i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (5.05 - 1.48i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (7.32 + 4.70i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (15.3 + 4.50i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.292 - 0.640i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 - 1.26T + 97T^{2} \) |
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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
−10.48558045961334678465358500858, −9.880299758859106477701250739122, −9.024081490699948704192694054187, −8.129769776779951472025003744772, −6.75712082571422455724330809500, −6.03289195816207161586650432023, −5.35136891975913737905795531342, −4.28449010674368129137980610796, −2.42010780508427374014069007294, −1.54714980545896343288089740145,
1.44003947377979930805874847537, 2.80378663631025801848607307313, 4.13638284783931352428131749917, 4.82281063075578746153606931294, 6.32332817221246157399953321738, 7.00874326749728474036685975759, 8.362568366955081364291169538144, 8.503810659100475725676517509718, 9.953336058062212776326376675008, 10.68549988989245638559270922451