L(s) = 1 | + (2.12 − 0.707i)5-s + (2 − 2i)7-s − 2.82i·11-s + (1 + i)13-s + (−2.82 − 2.82i)17-s + (−2.82 + 2.82i)23-s + (3.99 − 3i)25-s − 4.24·29-s + 4·31-s + (2.82 − 5.65i)35-s + (1 − i)37-s − 1.41i·41-s + (8 + 8i)43-s + (−5.65 − 5.65i)47-s − i·49-s + ⋯ |
L(s) = 1 | + (0.948 − 0.316i)5-s + (0.755 − 0.755i)7-s − 0.852i·11-s + (0.277 + 0.277i)13-s + (−0.685 − 0.685i)17-s + (−0.589 + 0.589i)23-s + (0.799 − 0.600i)25-s − 0.787·29-s + 0.718·31-s + (0.478 − 0.956i)35-s + (0.164 − 0.164i)37-s − 0.220i·41-s + (1.21 + 1.21i)43-s + (−0.825 − 0.825i)47-s − 0.142i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71515 - 0.773465i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71515 - 0.773465i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.12 + 0.707i)T \) |
good | 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-8 - 8i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.65 + 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−10.29124948767454297476766861920, −9.432897620136433802131283455339, −8.639871706981034757256281995726, −7.75824765978693877965497633000, −6.71315669860032674623267268912, −5.79325097986642771084822611564, −4.88876692996562072211791947370, −3.85984738747701830797880568521, −2.36239081526574953475322860758, −1.07964533813510318722554519203,
1.75485240687861788518654888601, 2.55774196438260814649600955485, 4.16528741659894803723803153654, 5.23350209337640075002975461239, 6.02755265316486403314834817555, 6.91518818403979801688156671763, 8.055032942059538705424759193060, 8.838020213833975394701134933421, 9.698413361719190999023869785375, 10.49782799633750316163330592902