| L(s) = 1 | + (−8.22 + 7.63i)2-s + (11.6 − 1.74i)3-s + (7.01 − 93.6i)4-s + (15.3 − 39.1i)5-s + (−82.1 + 102. i)6-s + (−87.9 + 95.2i)7-s + (433. + 543. i)8-s + (−100. + 31.0i)9-s + (172. + 438. i)10-s + (−193. − 59.8i)11-s + (−82.3 − 1.09e3i)12-s + (232. + 1.01e3i)13-s + (−3.07 − 1.45e3i)14-s + (109. − 480. i)15-s + (−4.73e3 − 714. i)16-s + (−594. + 405. i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 + 1.34i)2-s + (0.744 − 0.112i)3-s + (0.219 − 2.92i)4-s + (0.274 − 0.699i)5-s + (−0.931 + 1.16i)6-s + (−0.678 + 0.734i)7-s + (2.39 + 3.00i)8-s + (−0.413 + 0.127i)9-s + (0.544 + 1.38i)10-s + (−0.483 − 0.149i)11-s + (−0.165 − 2.20i)12-s + (0.380 + 1.66i)13-s + (−0.00419 − 1.98i)14-s + (0.125 − 0.551i)15-s + (−4.62 − 0.697i)16-s + (−0.498 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0283920 + 0.552816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0283920 + 0.552816i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (87.9 - 95.2i)T \) |
| good | 2 | \( 1 + (8.22 - 7.63i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-11.6 + 1.74i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-15.3 + 39.1i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (193. + 59.8i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (-232. - 1.01e3i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (594. - 405. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-524. - 908. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.21e3 - 831. i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (4.38e3 + 2.11e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (1.05e3 - 1.83e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-121. - 1.62e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (1.48e3 + 1.85e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (4.48e3 - 5.61e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (8.75e3 - 8.12e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (-56.4 + 753. i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (4.51e3 + 1.14e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (772. + 1.03e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-3.47e3 + 6.02e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-5.15e4 + 2.48e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-2.38e4 - 2.21e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (-3.39e4 - 5.87e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.95e4 + 8.57e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (1.28e5 - 3.95e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 1.38e4T + 8.58e9T^{2} \) |
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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
−15.44959622967209546048083774375, −14.39408933629948596365488871477, −13.37599397073887138298138307150, −11.22118190916958172277882488956, −9.488582857103777874430110467704, −9.020664739545354575389145408535, −8.080260668972035220975770487211, −6.59149757707442866406413441126, −5.38157205873863754816622898339, −1.83267169953339973127998843386,
0.40259354179590157287389224842, 2.64409829403794011815450997100, 3.42410227206581471116883817182, 7.11223479210123440392382884207, 8.251758932941165997131632934530, 9.417652806333679603097738975245, 10.38735847383743055148027941962, 11.11887411858600961655165112397, 12.78598307659921343170200512863, 13.56261208522424689023474026338
