L(s) = 1 | + (−3.36 + 5.83i)5-s + (−11.2 + 14.7i)7-s + (−26.8 − 46.4i)11-s + 1.73·13-s + (−23.4 − 40.6i)17-s + (10.0 − 17.4i)19-s + (59.4 − 103. i)23-s + (39.8 + 68.9i)25-s + 103.·29-s + (78.7 + 136. i)31-s + (−48.0 − 115. i)35-s + (18.8 − 32.6i)37-s + 287.·41-s + 504.·43-s + (−110. + 190. i)47-s + ⋯ |
L(s) = 1 | + (−0.301 + 0.521i)5-s + (−0.606 + 0.794i)7-s + (−0.735 − 1.27i)11-s + 0.0370·13-s + (−0.334 − 0.580i)17-s + (0.121 − 0.210i)19-s + (0.539 − 0.933i)23-s + (0.318 + 0.551i)25-s + 0.663·29-s + (0.456 + 0.790i)31-s + (−0.232 − 0.556i)35-s + (0.0838 − 0.145i)37-s + 1.09·41-s + 1.79·43-s + (−0.342 + 0.592i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.331667099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.331667099\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (11.2 - 14.7i)T \) |
good | 5 | \( 1 + (3.36 - 5.83i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (26.8 + 46.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 1.73T + 2.19e3T^{2} \) |
| 17 | \( 1 + (23.4 + 40.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10.0 + 17.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-59.4 + 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-78.7 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-18.8 + 32.6i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (110. - 190. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-146. - 253. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (297. + 516. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-132. + 229. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (468. + 811. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 545.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (149. + 259. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-470. + 814. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (588. - 1.01e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69783590675558426619184921408, −9.399179531290875639743785655827, −8.719791410761500828768695289473, −7.74095912023939143082649973880, −6.65869333546370496845752594104, −5.86445017541612854891432016055, −4.78025627880001454478427534093, −3.22325738652837513973249848797, −2.64264843154482831505231357713, −0.53631602909245864778121442302,
0.936868229129766546578690048265, 2.52589174722868294605492483958, 3.97369608578588513608872761678, 4.70850881330329908777157530769, 5.95972752126299226092516862454, 7.12168562849464462171296269616, 7.74556752150728592453067581795, 8.835933855188015080243330715947, 9.850282030518498551530573613229, 10.40854172488409887999603960938