L(s) = 1 | + (0.707 + 0.707i)5-s + 2.66i·7-s + (−3.49 − 3.49i)11-s + (2.94 − 2.94i)13-s − 1.85·17-s + (3.44 − 3.44i)19-s + 0.707i·23-s + 1.00i·25-s + (3.49 − 3.49i)29-s − 6.84·31-s + (−1.88 + 1.88i)35-s + (−0.0975 − 0.0975i)37-s − 10.2i·41-s + (−4.43 − 4.43i)43-s − 1.89·47-s + ⋯ |
L(s) = 1 | + (0.316 + 0.316i)5-s + 1.00i·7-s + (−1.05 − 1.05i)11-s + (0.815 − 0.815i)13-s − 0.448·17-s + (0.791 − 0.791i)19-s + 0.147i·23-s + 0.200i·25-s + (0.649 − 0.649i)29-s − 1.22·31-s + (−0.318 + 0.318i)35-s + (−0.0160 − 0.0160i)37-s − 1.59i·41-s + (−0.676 − 0.676i)43-s − 0.276·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.567217576\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.567217576\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 - 2.66iT - 7T^{2} \) |
| 11 | \( 1 + (3.49 + 3.49i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.94 + 2.94i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + (-3.44 + 3.44i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.707iT - 23T^{2} \) |
| 29 | \( 1 + (-3.49 + 3.49i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.84T + 31T^{2} \) |
| 37 | \( 1 + (0.0975 + 0.0975i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (4.43 + 4.43i)T + 43iT^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 + (-7.43 - 7.43i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.959 - 0.959i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.49 + 6.49i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.49 - 3.49i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.86iT - 71T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + (3.87 - 3.87i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−8.739943276680550372431998836085, −7.997342458769805823071415146091, −7.18709710160761429509654045557, −6.16611535064351811180295609279, −5.58865072269093378546015636483, −5.09155049365818994830687436879, −3.64102876981576148988092588169, −2.91239964141363927262962470481, −2.12805195010818537808234060510, −0.53477094156111581451136047807,
1.16907904457415940768201054528, 2.09383829628936834393632231352, 3.34419965671276174639832714973, 4.25613632247079671914850888954, 4.91871266243400673914976853408, 5.78532532457758476326514103332, 6.79461876436631377628187774324, 7.28835004432065617649963998972, 8.141087959254079472144870275381, 8.828966645490379683300975655239