L(s) = 1 | + (0.623 + 1.78i)2-s + (0.433 + 0.900i)3-s + (−2.00 + 1.59i)4-s + (−1.33 + 1.33i)6-s + (−2.49 − 1.57i)8-s + (−0.623 + 0.781i)9-s + (−2.30 − 1.11i)12-s + (−1.75 − 0.400i)13-s + (0.669 − 2.93i)16-s + (−1.78 − 0.623i)18-s + (0.433 + 0.900i)23-s + (0.330 − 2.93i)24-s + (0.623 + 0.781i)25-s + (−0.380 − 3.38i)26-s + (−0.974 − 0.222i)27-s + ⋯ |
L(s) = 1 | + (0.623 + 1.78i)2-s + (0.433 + 0.900i)3-s + (−2.00 + 1.59i)4-s + (−1.33 + 1.33i)6-s + (−2.49 − 1.57i)8-s + (−0.623 + 0.781i)9-s + (−2.30 − 1.11i)12-s + (−1.75 − 0.400i)13-s + (0.669 − 2.93i)16-s + (−1.78 − 0.623i)18-s + (0.433 + 0.900i)23-s + (0.330 − 2.93i)24-s + (0.623 + 0.781i)25-s + (−0.380 − 3.38i)26-s + (−0.974 − 0.222i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.224492877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224492877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.433 - 0.900i)T \) |
| 23 | \( 1 + (-0.433 - 0.900i)T \) |
| 29 | \( 1 + (0.781 - 0.623i)T \) |
good | 2 | \( 1 + (-0.623 - 1.78i)T + (-0.781 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 13 | \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 31 | \( 1 + (-1.33 + 0.467i)T + (0.781 - 0.623i)T^{2} \) |
| 37 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (0.752 - 0.752i)T - iT^{2} \) |
| 43 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 47 | \( 1 + (-1.05 - 1.68i)T + (-0.433 + 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 - 1.24iT - T^{2} \) |
| 61 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-1.87 - 0.656i)T + (0.781 + 0.623i)T^{2} \) |
| 79 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 97 | \( 1 + (0.974 + 0.222i)T^{2} \) |
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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
−9.481658734099226822035723487948, −9.043537416749684112895792750937, −8.052916756218446822845185075708, −7.56642689952843953414908251196, −6.84615865309500115865712514675, −5.74870096407138194464285945737, −5.07847042368699786486483923993, −4.61688692628966041352061418788, −3.59287744867740443558147174250, −2.75443183476180981967800331239,
0.66751816489813464279854681340, 2.10379207565987332566114184292, 2.51372762243530060108187544018, 3.48260005380094972863155331358, 4.53135710819732794873974862876, 5.19509665581139674737742817206, 6.34001055971070927083593078725, 7.19893971153842422921942951791, 8.304923947830623527281473904371, 9.008378920509109949471986378988