| L(s) = 1 | − 7i·3-s + (−2 + 11i)5-s − 7i·7-s − 22·9-s + 7·11-s + 3i·13-s + (77 + 14i)15-s − 61i·17-s + 48·19-s − 49·21-s − 58i·23-s + (−117 − 44i)25-s − 35i·27-s − 219·29-s − 298·31-s + ⋯ |
| L(s) = 1 | − 1.34i·3-s + (−0.178 + 0.983i)5-s − 0.377i·7-s − 0.814·9-s + 0.191·11-s + 0.0640i·13-s + (1.32 + 0.240i)15-s − 0.870i·17-s + 0.579·19-s − 0.509·21-s − 0.525i·23-s + (−0.936 − 0.351i)25-s − 0.249i·27-s − 1.40·29-s − 1.72·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6973866570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6973866570\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 - 11i)T \) |
| 7 | \( 1 + 7iT \) |
| good | 3 | \( 1 + 7iT - 27T^{2} \) |
| 11 | \( 1 - 7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 61iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 48T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 219T + 2.43e4T^{2} \) |
| 31 | \( 1 + 298T + 2.97e4T^{2} \) |
| 37 | \( 1 - 170iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 50T + 6.89e4T^{2} \) |
| 43 | \( 1 + 484iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 131iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 210iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 782T + 2.05e5T^{2} \) |
| 61 | \( 1 - 488T + 2.26e5T^{2} \) |
| 67 | \( 1 - 494iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 240T + 3.57e5T^{2} \) |
| 73 | \( 1 - 58iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 608T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3iT - 9.12e5T^{2} \) |
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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
−9.961042833648673140848314081562, −8.913141940762048963225530027985, −7.61253595456435756477548175425, −7.30363010184969756935119928104, −6.53308076459765528334895317156, −5.50528361542527529349233047467, −3.91683515470422372979935504591, −2.75448308566861511887732120854, −1.63168190880908975977603203653, −0.20319283788816358448575445104,
1.64210436803996131296962453551, 3.45866678975718679881030535214, 4.17730157787475113820873071645, 5.20534808390721976116086835452, 5.81685847453052346086832020536, 7.43447707162132302208371083089, 8.430323971688782385105555253646, 9.352813791767667303695912449028, 9.611664388159074821387749981107, 10.85740070706936975043669458075
