L(s) = 1 | − 9.89i·5-s + 30.9i·7-s − 43.8·11-s + 28·13-s − 49.4i·17-s − 43.8·23-s + 27·25-s − 137. i·29-s − 216. i·31-s + 306.·35-s − 266·37-s − 227. i·41-s + 61.9i·43-s + 306.·47-s − 617.·49-s + ⋯ |
L(s) = 1 | − 0.885i·5-s + 1.67i·7-s − 1.20·11-s + 0.597·13-s − 0.706i·17-s − 0.397·23-s + 0.215·25-s − 0.878i·29-s − 1.25i·31-s + 1.48·35-s − 1.18·37-s − 0.867i·41-s + 0.219i·43-s + 0.951·47-s − 1.79·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7932987649\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7932987649\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 9.89iT - 125T^{2} \) |
| 7 | \( 1 - 30.9iT - 343T^{2} \) |
| 11 | \( 1 + 43.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 43.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 137. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 216. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 266T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 61.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 120. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 350T + 2.26e5T^{2} \) |
| 67 | \( 1 + 433. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 657.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 112T + 3.89e5T^{2} \) |
| 79 | \( 1 - 216. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 306.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 616T + 9.12e5T^{2} \) |
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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
−9.847026422281096497927432293064, −8.982145878741116892480951548616, −8.450627282607873124022474725657, −7.53119635366343209086253951464, −6.01690580589182594729572899593, −5.45692881413857291213861421213, −4.54964073651829808912779227015, −2.96431693299985643827471125808, −1.96311358558347037278771411274, −0.23541224259061507130278464910,
1.34434535288140661233422688678, 2.97549654534338380511554071239, 3.83638272607287902164372165744, 4.98134802660682366656691333574, 6.27611453710054845904188439965, 7.11850860167752500922430068525, 7.75430381528797946993337129769, 8.776206266367528421211982694335, 10.27808980380094704333742230569, 10.49368728012644233793220443184