| L(s) = 1 | + (1.75 + 1.38i)5-s − 1.50·11-s + i·13-s − 2.72i·17-s − 0.726·19-s + 4.72i·23-s + (1.14 + 4.86i)25-s − 7.55·29-s − 3.00·31-s + 5.00i·37-s − 5.78·41-s + 2.72i·43-s − 10.2i·47-s + 7·49-s + 7.55i·53-s + ⋯ |
| L(s) = 1 | + (0.783 + 0.621i)5-s − 0.453·11-s + 0.277i·13-s − 0.661i·17-s − 0.166·19-s + 0.985i·23-s + (0.228 + 0.973i)25-s − 1.40·29-s − 0.540·31-s + 0.823i·37-s − 0.903·41-s + 0.415i·43-s − 1.49i·47-s + 49-s + 1.03i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098159397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098159397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.75 - 1.38i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 + 2.72iT - 17T^{2} \) |
| 19 | \( 1 + 0.726T + 19T^{2} \) |
| 23 | \( 1 - 4.72iT - 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 5.00iT - 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 43 | \( 1 - 2.72iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 7.55iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 + 7.78iT - 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.759461530740780467180565002304, −7.62723007256731898107850275040, −7.21348383142416693559525341412, −6.42591721172297187669787477953, −5.60021254918280908032851939676, −5.14440111942479451691307537530, −3.99632613537478789566895332304, −3.14935217996270840335666641041, −2.32376218571008333153546126415, −1.43977903962926244005598572791,
0.27553392848357052012657172700, 1.61642080019442197313070869995, 2.34458374553479956325734422953, 3.43492733731255321985367912248, 4.37472819897967762719263278948, 5.14038688791482945576165117922, 5.81549909634357433015035795755, 6.39946989738521429270424756177, 7.38362675912891075622090017153, 8.067280939904542458198515079497
