| L(s) = 1 | − 9·3-s + 76·5-s − 100·7-s + 81·9-s + 106·11-s − 169·13-s − 684·15-s + 234·17-s + 276·19-s + 900·21-s − 2.54e3·23-s + 2.65e3·25-s − 729·27-s + 8.26e3·29-s + 608·31-s − 954·33-s − 7.60e3·35-s − 2.01e3·37-s + 1.52e3·39-s + 8.84e3·41-s + 1.76e4·43-s + 6.15e3·45-s − 1.87e4·47-s − 6.80e3·49-s − 2.10e3·51-s − 2.69e4·53-s + 8.05e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.35·5-s − 0.771·7-s + 1/3·9-s + 0.264·11-s − 0.277·13-s − 0.784·15-s + 0.196·17-s + 0.175·19-s + 0.445·21-s − 1.00·23-s + 0.848·25-s − 0.192·27-s + 1.82·29-s + 0.113·31-s − 0.152·33-s − 1.04·35-s − 0.241·37-s + 0.160·39-s + 0.821·41-s + 1.45·43-s + 0.453·45-s − 1.23·47-s − 0.405·49-s − 0.113·51-s − 1.31·53-s + 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.071068541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.071068541\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 76 T + p^{5} T^{2} \) |
| 7 | \( 1 + 100 T + p^{5} T^{2} \) |
| 11 | \( 1 - 106 T + p^{5} T^{2} \) |
| 17 | \( 1 - 234 T + p^{5} T^{2} \) |
| 19 | \( 1 - 276 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2548 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8266 T + p^{5} T^{2} \) |
| 31 | \( 1 - 608 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2010 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8844 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17636 T + p^{5} T^{2} \) |
| 47 | \( 1 + 18770 T + p^{5} T^{2} \) |
| 53 | \( 1 + 26970 T + p^{5} T^{2} \) |
| 59 | \( 1 - 41966 T + p^{5} T^{2} \) |
| 61 | \( 1 - 778 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12632 T + p^{5} T^{2} \) |
| 71 | \( 1 + 40466 T + p^{5} T^{2} \) |
| 73 | \( 1 - 54302 T + p^{5} T^{2} \) |
| 79 | \( 1 - 44656 T + p^{5} T^{2} \) |
| 83 | \( 1 + 69918 T + p^{5} T^{2} \) |
| 89 | \( 1 + 44520 T + p^{5} T^{2} \) |
| 97 | \( 1 + 86026 T + p^{5} T^{2} \) |
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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
−9.871667909191998464250065004736, −9.308549862898110863814994841516, −8.099945882991428801004139340040, −6.81166258214305385414927266134, −6.21464578689606795192554021999, −5.49462800225665636351914547425, −4.40562519457017763375934548505, −3.01426348550849408038430859872, −1.91501641248954210564369766840, −0.71541225848001547287852013028,
0.71541225848001547287852013028, 1.91501641248954210564369766840, 3.01426348550849408038430859872, 4.40562519457017763375934548505, 5.49462800225665636351914547425, 6.21464578689606795192554021999, 6.81166258214305385414927266134, 8.099945882991428801004139340040, 9.308549862898110863814994841516, 9.871667909191998464250065004736
