| L(s) = 1 | + 3.27i·3-s − 4.21i·5-s − 0.547i·7-s − 7.75·9-s − 0.860i·11-s + 0.171·13-s + 13.8·15-s + (−3.68 − 1.84i)17-s + 0.806·19-s + 1.79·21-s + 1.57i·23-s − 12.7·25-s − 15.5i·27-s − 1.06i·29-s + 4.51i·31-s + ⋯ |
| L(s) = 1 | + 1.89i·3-s − 1.88i·5-s − 0.207i·7-s − 2.58·9-s − 0.259i·11-s + 0.0476·13-s + 3.57·15-s + (−0.894 − 0.447i)17-s + 0.185·19-s + 0.391·21-s + 0.327i·23-s − 2.55·25-s − 3.00i·27-s − 0.198i·29-s + 0.810i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7126996387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7126996387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.68 + 1.84i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.27iT - 3T^{2} \) |
| 5 | \( 1 + 4.21iT - 5T^{2} \) |
| 7 | \( 1 + 0.547iT - 7T^{2} \) |
| 11 | \( 1 + 0.860iT - 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 19 | \( 1 - 0.806T + 19T^{2} \) |
| 23 | \( 1 - 1.57iT - 23T^{2} \) |
| 29 | \( 1 + 1.06iT - 29T^{2} \) |
| 31 | \( 1 - 4.51iT - 31T^{2} \) |
| 37 | \( 1 - 8.01iT - 37T^{2} \) |
| 41 | \( 1 - 4.56iT - 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.00T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 61 | \( 1 - 4.06iT - 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.44iT - 71T^{2} \) |
| 73 | \( 1 + 5.64iT - 73T^{2} \) |
| 79 | \( 1 - 11.7iT - 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.06iT - 97T^{2} \) |
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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
−8.729974956432340462459637945403, −8.581587490197351629039870143763, −7.56890157694093875103023695591, −6.14932734438629148689108761404, −5.47184317686575748783504910789, −4.80177942525824003695900485465, −4.41188156531345966259380232429, −3.70649216037737671372966729119, −2.64382550951476106405400154665, −1.08405133318961673461588027292,
0.21691808854136536912533449450, 1.81570355540482641274848054329, 2.40499191877194683803706549883, 3.03441902754053243694345535988, 4.11271395555972842052476308163, 5.78023783587336897694154859000, 6.04624174775924660406499314604, 6.86128057784254082177809933879, 7.28088872452232739786875601770, 7.75360853053521199115489997043
