L(s) = 1 | + (−1.59 − 1.16i)2-s + (0.896 + 2.76i)4-s + (0.533 + 0.734i)5-s + (1.16 − 3.57i)8-s − 1.79i·10-s + (0.156 − 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (−1.54 + 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (−1.87 + 0.610i)26-s + 5.17·32-s + (3.24 − 1.05i)40-s + (−0.437 + 1.34i)41-s + ⋯ |
L(s) = 1 | + (−1.59 − 1.16i)2-s + (0.896 + 2.76i)4-s + (0.533 + 0.734i)5-s + (1.16 − 3.57i)8-s − 1.79i·10-s + (0.156 − 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (−1.54 + 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (−1.87 + 0.610i)26-s + 5.17·32-s + (3.24 − 1.05i)40-s + (−0.437 + 1.34i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5703491641\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5703491641\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.156 + 0.987i)T \) |
| 13 | \( 1 + (-0.587 + 0.809i)T \) |
good | 2 | \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.893081671995599104092371788028, −8.914045680330983540658754696614, −8.457174160912861038078164361818, −7.56353807162850211583288747174, −6.73313180351450643162329403311, −5.80933776318411123060605110988, −3.95205261976299428233217373905, −3.10417873952285042535252860305, −2.35073451998344739498823561280, −0.999076212008781096268224931793,
1.26012727978784758800376288542, 2.13701726387964526188768106087, 4.42970152119008982492506979317, 5.34721213924123730317942115531, 6.10952130737628475332599994960, 6.94444698675918518831893984650, 7.57425033446967410796846623841, 8.593341214875123791247105271074, 9.075189892482585387047379942537, 9.652703120877793628929443326828