L(s) = 1 | + 9·9-s − 24·13-s + 16·17-s + 42·29-s + 24·37-s − 18·41-s + 49·49-s + 56·53-s − 22·61-s − 96·73-s + 81·81-s + 78·89-s − 144·97-s + 198·101-s + 182·109-s + 224·113-s − 216·117-s + ⋯ |
L(s) = 1 | + 9-s − 1.84·13-s + 0.941·17-s + 1.44·29-s + 0.648·37-s − 0.439·41-s + 49-s + 1.05·53-s − 0.360·61-s − 1.31·73-s + 81-s + 0.876·89-s − 1.48·97-s + 1.96·101-s + 1.66·109-s + 1.98·113-s − 1.84·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.020032969\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.020032969\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 24 T + p^{2} T^{2} \) |
| 17 | \( 1 - 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 24 T + p^{2} T^{2} \) |
| 41 | \( 1 + 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 144 T + p^{2} T^{2} \) |
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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
−9.393042814151218165091305975053, −8.367947437203732592681296377908, −7.44135337971208069411033871903, −7.06906256904773512946803028043, −5.95531746637931879285004755844, −4.94864500131543919239892957295, −4.34684842009753130179988061886, −3.12336075728947567052362166040, −2.11480099809242131745707964976, −0.805396409916092753756013984338,
0.805396409916092753756013984338, 2.11480099809242131745707964976, 3.12336075728947567052362166040, 4.34684842009753130179988061886, 4.94864500131543919239892957295, 5.95531746637931879285004755844, 7.06906256904773512946803028043, 7.44135337971208069411033871903, 8.367947437203732592681296377908, 9.393042814151218165091305975053