| L(s) = 1 | + (−3.41 − 0.916i)2-s + (−5.16 − 8.94i)3-s + (−2.99 − 1.73i)4-s + (8.65 − 8.65i)5-s + (9.46 + 35.3i)6-s + (−2.94 + 0.790i)7-s + (48.7 + 48.7i)8-s + (−12.8 + 22.2i)9-s + (−37.5 + 21.6i)10-s + (53.6 − 200. i)11-s + 35.7i·12-s + (−168. − 7.63i)13-s + 10.8·14-s + (−122. − 32.7i)15-s + (−94.2 − 163. i)16-s + (419. + 242. i)17-s + ⋯ |
| L(s) = 1 | + (−0.854 − 0.229i)2-s + (−0.573 − 0.994i)3-s + (−0.187 − 0.108i)4-s + (0.346 − 0.346i)5-s + (0.262 + 0.981i)6-s + (−0.0601 + 0.0161i)7-s + (0.761 + 0.761i)8-s + (−0.158 + 0.274i)9-s + (−0.375 + 0.216i)10-s + (0.443 − 1.65i)11-s + 0.248i·12-s + (−0.998 − 0.0451i)13-s + 0.0551·14-s + (−0.542 − 0.145i)15-s + (−0.368 − 0.637i)16-s + (1.45 + 0.838i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.267245 - 0.503308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267245 - 0.503308i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (168. + 7.63i)T \) |
| good | 2 | \( 1 + (3.41 + 0.916i)T + (13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (5.16 + 8.94i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-8.65 + 8.65i)T - 625iT^{2} \) |
| 7 | \( 1 + (2.94 - 0.790i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-53.6 + 200. i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-419. - 242. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (4.81 + 17.9i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-100. + 57.7i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (59.3 + 102. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-1.05e3 + 1.05e3i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (-6.33 + 23.6i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-1.70e3 - 456. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-1.79e3 - 1.03e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (507. + 507. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + 2.09e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.55e3 + 415. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-1.64e3 + 2.84e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.18e3 + 316. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-263. - 983. i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-5.16e3 - 5.16e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 8.14e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (1.32e3 - 1.32e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (1.13e3 - 4.24e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (1.71e3 + 6.39e3i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−18.88444264611203363307155055206, −17.41206810418610591519754845317, −16.79163360692445180591895059559, −14.25744657288015328507636958416, −12.86748919621061419095158842000, −11.35106973608851923964047624380, −9.618012604746936781699663043543, −7.971080861895525401572519320934, −5.83124350202428669737781814932, −0.964022301098872677736570105978,
4.70426779878909223357889109756, 7.32807006469387185852424653131, 9.618955056574643509555883938244, 10.18268236521975063162065861883, 12.28361333922570638413964644526, 14.40999067552919611163806787564, 15.95985607867702915888603218531, 17.08350603036188924915008558495, 17.89473427570735481792369737160, 19.39442432115643758141916091157
