L(s) = 1 | + (−3.40 − 3.40i)5-s − 12.1·7-s + (−9.81 + 9.81i)11-s + (7.76 − 7.76i)13-s − 9.73·17-s + (11.2 + 11.2i)19-s − 20.2·23-s − 1.80i·25-s + (−16.4 + 16.4i)29-s − 26.3i·31-s + (41.3 + 41.3i)35-s + (23.7 + 23.7i)37-s − 24.7i·41-s + (29.8 − 29.8i)43-s + 31.3i·47-s + ⋯ |
L(s) = 1 | + (−0.681 − 0.681i)5-s − 1.73·7-s + (−0.891 + 0.891i)11-s + (0.597 − 0.597i)13-s − 0.572·17-s + (0.593 + 0.593i)19-s − 0.881·23-s − 0.0720i·25-s + (−0.565 + 0.565i)29-s − 0.850i·31-s + (1.18 + 1.18i)35-s + (0.641 + 0.641i)37-s − 0.603i·41-s + (0.694 − 0.694i)43-s + 0.666i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8028124174\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8028124174\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (3.40 + 3.40i)T + 25iT^{2} \) |
| 7 | \( 1 + 12.1T + 49T^{2} \) |
| 11 | \( 1 + (9.81 - 9.81i)T - 121iT^{2} \) |
| 13 | \( 1 + (-7.76 + 7.76i)T - 169iT^{2} \) |
| 17 | \( 1 + 9.73T + 289T^{2} \) |
| 19 | \( 1 + (-11.2 - 11.2i)T + 361iT^{2} \) |
| 23 | \( 1 + 20.2T + 529T^{2} \) |
| 29 | \( 1 + (16.4 - 16.4i)T - 841iT^{2} \) |
| 31 | \( 1 + 26.3iT - 961T^{2} \) |
| 37 | \( 1 + (-23.7 - 23.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 24.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-29.8 + 29.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 31.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-36.8 - 36.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-14.1 + 14.1i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-42.5 + 42.5i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-48.7 - 48.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 7.73T + 5.04e3T^{2} \) |
| 73 | \( 1 + 85.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-62.1 - 62.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 147.T + 9.40e3T^{2} \) |
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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
−9.736457650281133199574524196825, −8.829843635728537131140075543962, −7.950212041035849567617279618092, −7.24449652256049248933373358331, −6.22197183870752648862445730446, −5.44688434501306570366322728723, −4.25675186618451300176969723633, −3.50758688112491268365620305961, −2.38226834100049721041726725790, −0.59158231886506528181281464351,
0.44219321575892394956810785108, 2.53767192888790771217542594283, 3.34510679278803863733326762545, 4.03197130327676134203655552101, 5.52909960974371048963438735502, 6.35283510826375768705450930202, 6.99357398144916947501041140232, 7.86267757152250602012549456472, 8.845378741696849054419224027779, 9.590344134056261923774033238399