L(s) = 1 | + 3.21i·5-s + 1.36i·11-s + 2.93i·13-s + 6.91i·17-s − 7.35·19-s + 3.62i·23-s − 5.33·25-s + 1.11·29-s − 8.70·31-s + 7.63·37-s − 0.833i·41-s + 4.82i·43-s + 2.95·47-s + 4.57·53-s − 4.39·55-s + ⋯ |
L(s) = 1 | + 1.43i·5-s + 0.412i·11-s + 0.812i·13-s + 1.67i·17-s − 1.68·19-s + 0.756i·23-s − 1.06·25-s + 0.206·29-s − 1.56·31-s + 1.25·37-s − 0.130i·41-s + 0.735i·43-s + 0.430·47-s + 0.628·53-s − 0.592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8928180255\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8928180255\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.21iT - 5T^{2} \) |
| 11 | \( 1 - 1.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 6.91iT - 17T^{2} \) |
| 19 | \( 1 + 7.35T + 19T^{2} \) |
| 23 | \( 1 - 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 + 0.833iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 6.49iT - 73T^{2} \) |
| 79 | \( 1 - 7.79iT - 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 - 12.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.183392252499578751428327669271, −7.63503600795211668179085006268, −6.84068808346347437158817189090, −6.34227259778825040035283727981, −5.86205603792677419190911316898, −4.63659662370893208533643759286, −3.94203558595023277165611037852, −3.32222149303470947376213610663, −2.22892581501558591715071492479, −1.74764038244705617405733688838,
0.23691610720906243972611928420, 0.966840967612186044873720634701, 2.16885156746215684565884441397, 3.00080467488892753469436342848, 4.15322787145109997555542043536, 4.61371979818428335536242943033, 5.42189935069954333960149723255, 5.90254193277536214209752271203, 6.88154608240927246333663983703, 7.63772633244298023993963653636