L(s) = 1 | + (−0.456 − 1.33i)2-s + (0.707 + 0.707i)3-s + (−1.58 + 1.22i)4-s + (−1.50 + 1.65i)5-s + (0.623 − 1.26i)6-s − 2.58·7-s + (2.35 + 1.55i)8-s + 1.00i·9-s + (2.90 + 1.25i)10-s + (4.39 + 4.39i)11-s + (−1.98 − 0.254i)12-s + (−0.417 − 0.417i)13-s + (1.18 + 3.46i)14-s + (−2.23 + 0.110i)15-s + (1.00 − 3.87i)16-s + 4.40i·17-s + ⋯ |
L(s) = 1 | + (−0.323 − 0.946i)2-s + (0.408 + 0.408i)3-s + (−0.791 + 0.611i)4-s + (−0.671 + 0.741i)5-s + (0.254 − 0.518i)6-s − 0.978·7-s + (0.834 + 0.551i)8-s + 0.333i·9-s + (0.918 + 0.395i)10-s + (1.32 + 1.32i)11-s + (−0.572 − 0.0734i)12-s + (−0.115 − 0.115i)13-s + (0.316 + 0.926i)14-s + (−0.576 + 0.0286i)15-s + (0.252 − 0.967i)16-s + 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.716955 + 0.379040i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.716955 + 0.379040i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.456 + 1.33i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.50 - 1.65i)T \) |
good | 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 + (-4.39 - 4.39i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.417 + 0.417i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (4.53 - 4.53i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.281T + 23T^{2} \) |
| 29 | \( 1 + (-3.73 + 3.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + (-5.26 + 5.26i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.16iT - 41T^{2} \) |
| 43 | \( 1 + (-2.66 + 2.66i)T - 43iT^{2} \) |
| 47 | \( 1 - 7.45iT - 47T^{2} \) |
| 53 | \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.60 + 4.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.211 - 0.211i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.17 - 7.17i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 + (-4.56 - 4.56i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6.78iT - 97T^{2} \) |
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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
−12.36072032203765928160322279548, −11.20038784335195152957701940570, −10.22889614166267989961995094036, −9.682861212770318812217822202329, −8.612580690284722259535051153910, −7.52999826162623309115043540000, −6.36444109792255052970443334033, −4.18005816002601969519214869621, −3.72141758760095844161416353999, −2.22234052887567904627261395996,
0.72476655195163809963581580531, 3.43964224141699094321091854873, 4.71280795157079031701521629158, 6.23709422659525933150675808851, 6.90639144125351870468599375507, 8.122560944717573061788216467123, 8.993233545281960518368360337776, 9.416332246094212436985599812576, 11.05028062252650333185936759534, 12.14492850436294897448768288217