| 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
−9.113871950620968120759849601851, −8.725434188318409098360146807871, −7.59020850514502973561555626911, −6.82831561822013082708047393265, −6.29091755807794665206228127158, −5.21136402672063117024022424190, −4.35118357212342791921834186669, −2.92375189151499788196178862860, −1.82032635339650145079287179848, −0.40257332936735808558594623100,
1.56978298991214834476350421410, 3.18393296070522208788095908421, 4.16793754532892976921673650028, 4.75403945014907289465644059732, 6.08549574358512315846666321331, 6.38493409967277455843294220206, 7.65984249486020069343304262616, 8.653578387616798356928954837321, 9.470542002452704331338617786882, 9.899354768411074914727259846069
