| L(s) = 1 | + (3 − 4i)5-s + (7 + 7i)7-s + 10·11-s + (9 − 9i)13-s + (−1 − i)17-s + 8i·19-s + (−23 + 23i)23-s + (−7 − 24i)25-s + 8i·29-s + 14·31-s + (49 − 7i)35-s + (33 + 33i)37-s + 14·41-s + (15 − 15i)43-s + (−39 − 39i)47-s + ⋯ |
| L(s) = 1 | + (0.600 − 0.800i)5-s + (1 + i)7-s + 0.909·11-s + (0.692 − 0.692i)13-s + (−0.0588 − 0.0588i)17-s + 0.421i·19-s + (−1 + i)23-s + (−0.280 − 0.959i)25-s + 0.275i·29-s + 0.451·31-s + (1.39 − 0.199i)35-s + (0.891 + 0.891i)37-s + 0.341·41-s + (0.348 − 0.348i)43-s + (−0.829 − 0.829i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.513608501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.513608501\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3 + 4i)T \) |
| good | 7 | \( 1 + (-7 - 7i)T + 49iT^{2} \) |
| 11 | \( 1 - 10T + 121T^{2} \) |
| 13 | \( 1 + (-9 + 9i)T - 169iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (23 - 23i)T - 529iT^{2} \) |
| 29 | \( 1 - 8iT - 841T^{2} \) |
| 31 | \( 1 - 14T + 961T^{2} \) |
| 37 | \( 1 + (-33 - 33i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-15 + 15i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (39 + 39i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 56iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 98T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49 + 49i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 96iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63 - 63i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 112iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33 - 33i)T + 9.40e3iT^{2} \) |
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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
−10.01419838776438745095840123690, −9.290673215545744174008484908197, −8.430617383784460907522646142375, −7.969322647677959939629691636076, −6.40530068157379139860567763140, −5.65540058920006743653805297383, −4.91599197486204087057238899903, −3.72654279820159866866042886056, −2.14808219076491784280771570545, −1.20153402583324936702428003609,
1.17087442095904536008962883593, 2.32025632687975471213759164623, 3.84683340037927167750113122166, 4.53429292528685720498073819193, 5.98657338667022339334989494905, 6.66324816031926433044379137480, 7.52843610390515647406408300019, 8.479797822870693464488944634349, 9.475012233605069160493288152841, 10.28620738372544710916782614417
