L(s) = 1 | − 1.41·2-s + (2.05 − 2.18i)3-s + 2.00·4-s + (−2.90 + 3.08i)6-s + 2.64i·7-s − 2.82·8-s + (−0.539 − 8.98i)9-s + 10.4i·11-s + (4.11 − 4.36i)12-s + 4.25i·13-s − 3.74i·14-s + 4.00·16-s − 3.70·17-s + (0.763 + 12.7i)18-s + 7.95·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.685 − 0.727i)3-s + 0.500·4-s + (−0.484 + 0.514i)6-s + 0.377i·7-s − 0.353·8-s + (−0.0599 − 0.998i)9-s + 0.954i·11-s + (0.342 − 0.363i)12-s + 0.327i·13-s − 0.267i·14-s + 0.250·16-s − 0.217·17-s + (0.0423 + 0.705i)18-s + 0.418·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.624257747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624257747\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-2.05 + 2.18i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 4.25iT - 169T^{2} \) |
| 17 | \( 1 + 3.70T + 289T^{2} \) |
| 19 | \( 1 - 7.95T + 361T^{2} \) |
| 23 | \( 1 + 2.84T + 529T^{2} \) |
| 29 | \( 1 - 43.7iT - 841T^{2} \) |
| 31 | \( 1 - 49.6T + 961T^{2} \) |
| 37 | \( 1 - 4.01iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.84iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 1.10T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 91.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 117. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 29.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 108.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 134. iT - 9.40e3T^{2} \) |
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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
−9.644150989637327902581296996212, −8.807363159790508568857000550528, −8.264988070761154445694347765953, −7.22956127045311377848348896759, −6.82123087433140460503920878515, −5.72867592859050401554032175695, −4.41271096078455666634125676682, −3.07896923744392751618723994711, −2.17315014434424790954310000476, −1.12891944061745972137086337929,
0.66169346930185573574015742695, 2.28226763768477345370520591693, 3.26405493961946026451696794630, 4.20321318360698370361815639039, 5.37145823873746032428851072839, 6.37062757044703802026382741656, 7.50084322429232357615835129836, 8.236207588792410821644030646294, 8.760344615249634771251590410140, 9.814420618976971793342124272794