L(s) = 1 | + (−0.707 − 0.707i)5-s + 0.982i·7-s + (−1.62 − 1.62i)11-s + (−0.690 + 0.690i)13-s + 2.19·17-s + (−1.92 + 1.92i)19-s + 2.01i·23-s + 1.00i·25-s + (5.27 − 5.27i)29-s − 0.435·31-s + (0.694 − 0.694i)35-s + (−5.79 − 5.79i)37-s − 3.93i·41-s + (0.507 + 0.507i)43-s − 9.21·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s + 0.371i·7-s + (−0.490 − 0.490i)11-s + (−0.191 + 0.191i)13-s + 0.532·17-s + (−0.441 + 0.441i)19-s + 0.420i·23-s + 0.200i·25-s + (0.978 − 0.978i)29-s − 0.0781·31-s + (0.117 − 0.117i)35-s + (−0.953 − 0.953i)37-s − 0.613i·41-s + (0.0774 + 0.0774i)43-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5172565585\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5172565585\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.982iT - 7T^{2} \) |
| 11 | \( 1 + (1.62 + 1.62i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.690 - 0.690i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 + (1.92 - 1.92i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.01iT - 23T^{2} \) |
| 29 | \( 1 + (-5.27 + 5.27i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.435T + 31T^{2} \) |
| 37 | \( 1 + (5.79 + 5.79i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.93iT - 41T^{2} \) |
| 43 | \( 1 + (-0.507 - 0.507i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + (6.29 + 6.29i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.67 + 5.67i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.60 - 3.60i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.53 - 4.53i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 9.24iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (0.683 - 0.683i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.44iT - 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−8.406157911877886205905429126990, −7.85751520230052651923243546733, −7.02085668576374601293464473575, −6.05215317214412731934158892083, −5.42990279897675829397923914956, −4.56292107865595533164748823131, −3.65705755036501504266515697285, −2.74557644564981285216495828403, −1.62320508269943653253101785899, −0.16524832122426945115804938172,
1.35998522584803139424480528503, 2.66561368237785437698024855945, 3.37683365097489214170257752658, 4.51529291237015168826421948786, 5.02571886314518526236831406269, 6.16575013181605234647047515511, 6.87985044584987047678229604000, 7.56006825381472683874803587588, 8.240037472200697985540048339299, 9.012625345012305158277836438872