L(s) = 1 | + (1.16 − 0.672i)2-s + (−0.0962 + 0.166i)4-s + (0.868 + 2.06i)5-s + (−3.39 − 1.96i)7-s + 2.94i·8-s + (2.39 + 1.81i)10-s + (1.37 + 2.38i)11-s + (1.14 + 3.41i)13-s − 5.27·14-s + (1.78 + 3.09i)16-s + (4.09 + 2.36i)17-s + (1.85 − 3.20i)19-s + (−0.427 − 0.0536i)20-s + (3.20 + 1.84i)22-s + (−4.57 + 2.64i)23-s + ⋯ |
L(s) = 1 | + (0.823 − 0.475i)2-s + (−0.0481 + 0.0833i)4-s + (0.388 + 0.921i)5-s + (−1.28 − 0.741i)7-s + 1.04i·8-s + (0.757 + 0.574i)10-s + (0.414 + 0.718i)11-s + (0.318 + 0.947i)13-s − 1.40·14-s + (0.447 + 0.774i)16-s + (0.992 + 0.573i)17-s + (0.424 − 0.735i)19-s + (−0.0955 − 0.0119i)20-s + (0.683 + 0.394i)22-s + (−0.953 + 0.550i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67679 + 0.906219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67679 + 0.906219i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.868 - 2.06i)T \) |
| 13 | \( 1 + (-1.14 - 3.41i)T \) |
good | 2 | \( 1 + (-1.16 + 0.672i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (3.39 + 1.96i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.37 - 2.38i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.09 - 2.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.85 + 3.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.57 - 2.64i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.31 - 2.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + (3.66 - 2.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.408 - 0.708i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.85 + 2.80i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8.36iT - 47T^{2} \) |
| 53 | \( 1 + 7.01iT - 53T^{2} \) |
| 59 | \( 1 + (1.09 - 1.89i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.41 + 11.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 + 6.28i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.08 + 10.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 0.955iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 11.7iT - 83T^{2} \) |
| 89 | \( 1 + (-5.60 - 9.70i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.52 - 0.882i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.92279478916244729078415271687, −9.995475542051902113160401698968, −9.475985697708077090528636173955, −8.095570303338025643185953230192, −6.92767073811986488044097391015, −6.42341312264200061410844693721, −5.14386955690134779193149288520, −3.78890921874105593535018941610, −3.40340763134690460815730813668, −2.03423229776742785247903771917,
0.863422828009813118834504242209, 2.98400283177605228232531565170, 3.98381830389147740610943256632, 5.30843369226159918633240159052, 5.86236815381081192275202037397, 6.43718540103397948428179795630, 7.914894537292257753403882890984, 8.872886972016935644498383730596, 9.769060783763649989003833352229, 10.17928053180761439930412759501