L(s) = 1 | + (−32 − 32i)2-s + (748. − 748. i)3-s + 2.04e3i·4-s − 4.79e4·6-s + (1.21e5 + 1.21e5i)7-s + (6.55e4 − 6.55e4i)8-s − 5.89e5i·9-s − 8.55e5·11-s + (1.53e6 + 1.53e6i)12-s + (−5.63e5 + 5.63e5i)13-s − 7.74e6i·14-s − 4.19e6·16-s + (1.33e7 + 1.33e7i)17-s + (−1.88e7 + 1.88e7i)18-s + 9.07e7i·19-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (1.02 − 1.02i)3-s + 0.5i·4-s − 1.02·6-s + (1.02 + 1.02i)7-s + (0.250 − 0.250i)8-s − 1.10i·9-s − 0.482·11-s + (0.513 + 0.513i)12-s + (−0.116 + 0.116i)13-s − 1.02i·14-s − 0.250·16-s + (0.554 + 0.554i)17-s + (−0.554 + 0.554i)18-s + 1.92i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.178168958\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.178168958\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (32 + 32i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-748. + 748. i)T - 5.31e5iT^{2} \) |
| 7 | \( 1 + (-1.21e5 - 1.21e5i)T + 1.38e10iT^{2} \) |
| 11 | \( 1 + 8.55e5T + 3.13e12T^{2} \) |
| 13 | \( 1 + (5.63e5 - 5.63e5i)T - 2.32e13iT^{2} \) |
| 17 | \( 1 + (-1.33e7 - 1.33e7i)T + 5.82e14iT^{2} \) |
| 19 | \( 1 - 9.07e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + (6.22e7 - 6.22e7i)T - 2.19e16iT^{2} \) |
| 29 | \( 1 - 9.61e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 + 1.55e9T + 7.87e17T^{2} \) |
| 37 | \( 1 + (7.41e8 + 7.41e8i)T + 6.58e18iT^{2} \) |
| 41 | \( 1 - 9.28e8T + 2.25e19T^{2} \) |
| 43 | \( 1 + (-2.23e9 + 2.23e9i)T - 3.99e19iT^{2} \) |
| 47 | \( 1 + (-7.22e9 - 7.22e9i)T + 1.16e20iT^{2} \) |
| 53 | \( 1 + (-9.05e8 + 9.05e8i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 - 1.35e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 6.94e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + (1.13e11 + 1.13e11i)T + 8.18e21iT^{2} \) |
| 71 | \( 1 - 1.09e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-8.58e9 + 8.58e9i)T - 2.29e22iT^{2} \) |
| 79 | \( 1 + 1.74e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (-2.87e11 + 2.87e11i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 - 2.51e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (2.97e11 + 2.97e11i)T + 6.93e23iT^{2} \) |
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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
−12.70550418866276725504916807905, −12.12825664619157294149496972073, −10.66567793021022017095217870401, −9.096671497513203932125217266732, −8.198147011965248916043734422842, −7.49998638951817758642643491568, −5.57567763366499733375388040887, −3.45534061250252217518276201118, −2.05853611807583513964056498902, −1.50865961299634322263185184718,
0.60923654570945186390706323494, 2.44205169695443463157956886411, 4.07477294724975816556806974815, 5.10228617993318397162240225926, 7.24398742836646459552438068066, 8.171388218007912678188698065969, 9.274772527696183791956683801634, 10.28553978757166963442562503004, 11.25197061116816448580576163400, 13.46662400594209811033884839541