L(s) = 1 | + (22.6 − 39.1i)2-s + (−444. + 256. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−1.26e4 − 7.32e3i)5-s + 2.32e4i·6-s + (1.17e5 + 1.00e4i)7-s − 9.26e4·8-s + (−1.34e5 + 2.32e5i)9-s + (−5.74e5 + 3.31e5i)10-s + (1.34e6 + 2.32e6i)11-s + (9.09e5 + 5.25e5i)12-s + 1.34e6i·13-s + (3.04e6 − 4.36e6i)14-s + 7.51e6·15-s + (−2.09e6 + 3.63e6i)16-s + (6.20e6 − 3.58e6i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.609 + 0.351i)3-s + (−0.249 − 0.433i)4-s + (−0.812 − 0.468i)5-s + 0.497i·6-s + (0.996 + 0.0856i)7-s − 0.353·8-s + (−0.252 + 0.437i)9-s + (−0.574 + 0.331i)10-s + (0.757 + 1.31i)11-s + (0.304 + 0.175i)12-s + 0.279i·13-s + (0.404 − 0.579i)14-s + 0.659·15-s + (−0.125 + 0.216i)16-s + (0.257 − 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(1.17509 + 0.521041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17509 + 0.521041i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-22.6 + 39.1i)T \) |
| 7 | \( 1 + (-1.17e5 - 1.00e4i)T \) |
good | 3 | \( 1 + (444. - 256. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (1.26e4 + 7.32e3i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-1.34e6 - 2.32e6i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 - 1.34e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-6.20e6 + 3.58e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-6.33e7 - 3.65e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (1.14e8 - 1.97e8i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + 5.69e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-2.67e8 + 1.54e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (8.51e8 - 1.47e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 5.37e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 9.19e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-9.45e9 - 5.45e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-7.92e9 - 1.37e10i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-1.75e10 + 1.01e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-2.38e10 - 1.37e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (3.35e10 + 5.80e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 - 7.34e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-5.62e10 + 3.24e10i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (3.16e9 - 5.48e9i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 5.07e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (5.49e11 + 3.17e11i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 3.89e11iT - 6.93e23T^{2} \) |
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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
−16.81348289688186080467984555935, −15.37809350908136093665801130371, −14.01149385081027955602921276773, −11.99851937962303309044763305283, −11.53652453288330794016303886867, −9.793997132213113051491209893362, −7.82523628165505248660998736095, −5.27585349128594635188636906950, −4.12956639619982676632450718646, −1.54622848451045528472781611245,
0.57856515117868350209290263706, 3.61651652465282917222708105404, 5.57904678063004349793684619832, 7.09053459647901518665976107502, 8.533803173099343437861328245308, 11.19865481702059343530300908520, 11.98360459103837748423439983760, 13.93280358696601763610782865323, 15.00417061522099874080235426608, 16.44289039748215421697018207769