L(s) = 1 | + 2.00·3-s + (1.06 + 0.770i)5-s + (0.823 + 2.53i)7-s + 1.02·9-s + (−0.383 + 0.278i)11-s + (0.477 − 1.46i)13-s + (2.12 + 1.54i)15-s + (4.19 − 3.04i)17-s + (1.37 + 4.24i)19-s + (1.65 + 5.09i)21-s + (−2.17 + 6.70i)23-s + (−1.01 − 3.12i)25-s − 3.95·27-s + (−0.481 − 0.349i)29-s + (2.56 − 1.86i)31-s + ⋯ |
L(s) = 1 | + 1.15·3-s + (0.474 + 0.344i)5-s + (0.311 + 0.958i)7-s + 0.343·9-s + (−0.115 + 0.0840i)11-s + (0.132 − 0.407i)13-s + (0.549 + 0.399i)15-s + (1.01 − 0.739i)17-s + (0.316 + 0.974i)19-s + (0.360 + 1.11i)21-s + (−0.454 + 1.39i)23-s + (−0.202 − 0.624i)25-s − 0.761·27-s + (−0.0893 − 0.0649i)29-s + (0.460 − 0.334i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.28009 + 0.671082i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.28009 + 0.671082i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 41 | \( 1 + (5.33 + 3.53i)T \) |
good | 3 | \( 1 - 2.00T + 3T^{2} \) |
| 5 | \( 1 + (-1.06 - 0.770i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.823 - 2.53i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.383 - 0.278i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.477 + 1.46i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.19 + 3.04i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 4.24i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.17 - 6.70i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.481 + 0.349i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.56 + 1.86i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.06 - 0.772i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 6.48i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.404 + 1.24i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.60 - 1.16i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.42 + 7.47i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.08 - 6.40i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.25 + 6.72i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (3.24 - 2.35i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + 3.85T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + (3.69 + 11.3i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.72 - 6.33i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.27884021570698515982823689285, −9.702333981710284031022303620905, −8.862150198989720743362890351595, −8.059340438532341258675105034117, −7.41755902102622946951815223112, −5.93896716722826970927431020586, −5.35106806459675107083428887504, −3.70741597715712085672526701938, −2.81619448932065264304496781361, −1.86200354267476556879475737439,
1.33473270280594034048871027614, 2.66775182325731964990032225277, 3.76772776395661816892321677162, 4.75290041157894613727901348233, 5.99345093472929161644191713173, 7.14346956535540778764047573814, 7.997295143386837046661620797071, 8.669205873725404800911395517990, 9.521242957239248182997579516517, 10.28391287379285322478477074939