L(s) = 1 | + (−0.866 + 1.5i)3-s + (−1.5 − 2.59i)9-s + 3·11-s + 2i·13-s − 5.19·17-s + 5.19i·19-s + 6i·23-s + 5.19·27-s − 10.3i·29-s + 3.46i·31-s + (−2.59 + 4.5i)33-s + 8i·37-s + (−3 − 1.73i)39-s + 5.19i·41-s + 3.46·43-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (−0.5 − 0.866i)9-s + 0.904·11-s + 0.554i·13-s − 1.26·17-s + 1.19i·19-s + 1.25i·23-s + 1.00·27-s − 1.92i·29-s + 0.622i·31-s + (−0.452 + 0.783i)33-s + 1.31i·37-s + (−0.480 − 0.277i)39-s + 0.811i·41-s + 0.528·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8771818751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8771818751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 5.19iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.899354768411074914727259846069, −9.470542002452704331338617786882, −8.653578387616798356928954837321, −7.65984249486020069343304262616, −6.38493409967277455843294220206, −6.08549574358512315846666321331, −4.75403945014907289465644059732, −4.16793754532892976921673650028, −3.18393296070522208788095908421, −1.56978298991214834476350421410,
0.40257332936735808558594623100, 1.82032635339650145079287179848, 2.92375189151499788196178862860, 4.35118357212342791921834186669, 5.21136402672063117024022424190, 6.29091755807794665206228127158, 6.82831561822013082708047393265, 7.59020850514502973561555626911, 8.725434188318409098360146807871, 9.113871950620968120759849601851